In
mathematicsMathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a ring is an
algebraic structureIn abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
consisting of a set together with two binary operations usually called
additionAddition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
and
multiplicationMultiplication is the mathematical operation of scaling one number by another. It is one of the four basic operations in elementary arithmetic ....
, where the set is an
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under addition (called the additive group of the ring) and a
semigroupIn mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation. A semigroup generalizes a monoid in that there might not exist an identity element...
under multiplication such that multiplication distributes over addition. In other words the ring
axiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be selfevident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
s require that addition is
commutativeIn mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
, addition and multiplication are
associativeIn mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
, multiplication
distributesIn mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
over addition, each element in the set has an
additive inverseIn mathematics, the additive inverse, or opposite, of a number a is the number that, when added to a, yields zero.The additive inverse of a is denoted −a....
, and there exists an
additive identityIn mathematics the additive identity of a set which is equipped with the operation of addition is an element which, when added to any element x in the set, yields x...
. One of the most common examples of a ring is the set of integers endowed with its natural operations of addition and multiplication. Certain variations of the definition of a ring are sometimes employed, and these are outlined later in the article.
The branch of mathematics that studies rings is known as
ring theoryIn abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
. Ring theorists study properties common to both familiar
mathematical structureIn mathematics, a structure on a set, or more generally a type, consists of additional mathematical objects that in some manner attach to the set, making it easier to visualize or work with, or endowing the collection with meaning or significance....
s such as
integerThe integers are formed by the natural numbers together with the negatives of the nonzero natural numbers .They are known as Positive and Negative Integers respectively...
s and
polynomialIn mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and nonnegative integer exponents...
s, and to the many less wellknown mathematical structures that also satisfy the axioms of ring theory. The ubiquity of rings makes them a central organizing principle of contemporary mathematics.
Ring theory may be used to understand fundamental
physical lawA physical law or scientific law is "a theoretical principle deduced from particular facts, applicable to a defined group or class of phenomena, and expressible by the statement that a particular phenomenon always occurs if certain conditions be present." Physical laws are typically conclusions...
s, such as those underlying
special relativitySpecial relativity is the physical theory of measurement in an inertial frame of reference proposed in 1905 by Albert Einstein in the paper "On the Electrodynamics of Moving Bodies".It generalizes Galileo's...
and symmetry phenomena in molecular
chemistryChemistry is the science of matter, especially its chemical reactions, but also its composition, structure and properties. Chemistry is concerned with atoms and their interactions with other atoms, and particularly with the properties of chemical bonds....
.
The concept of a ring first arose from attempts to prove
Fermat's last theoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
, starting with
Richard DedekindJulius Wilhelm Richard Dedekind was a German mathematician who did important work in abstract algebra , algebraic number theory and the foundations of the real numbers.Life:...
in the 1880s. After contributions from other fields, mainly
number theoryNumber theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, the ring notion was generalized and firmly established during the 1920s by
Emmy NoetherAmalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
and
Wolfgang KrullWolfgang Krull was a German mathematician working in the field of commutative algebra.He was born in BadenBaden, Imperial Germany and died in Bonn, West Germany. See also :* Krull dimension* Krull topology...
. Modern ring theory—a very active mathematical discipline—studies rings in their own right. To explore rings, mathematicians have devised
various notionsRing theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.Definition of a ring:...
to break rings into smaller, betterunderstandable pieces, such as
idealIn ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
s,
quotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
s and
simple ringIn abstract algebra, a simple ring is a nonzero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
s. In addition to these abstract properties, ring theorists also make various distinctions between the theory of
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s and
noncommutative ringIn mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.Noncommutative rings are ubiquitous in mathematics, and occur...
s—the former belonging to
algebraic number theoryAlgebraic number theory is a major branch of number theory which studies algebraic structures related to algebraic integers. This is generally accomplished by considering a ring of algebraic integers O in an algebraic number field K/Q, and studying their algebraic properties such as factorization,...
and
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
. A particularly rich theory has been developed for a certain special class of commutative rings, known as
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
s, which lies within the realm of
field theoryField theory is a branch of mathematics which studies the properties of fields. A field is a mathematical entity for which addition, subtraction, multiplication and division are welldefined....
. Likewise, the corresponding theory for noncommutative rings, that of noncommutative
division ringIn abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with...
s, constitutes an active research interest for noncommutative ring theorists. Since the discovery of a mysterious connection between noncommutative ring theory and geometry during the 1980s by
Alain ConnesAlain Connes is a French mathematician, currently Professor at the Collège de France, IHÉS, The Ohio State University and Vanderbilt University.Work:...
,
noncommutative geometryNoncommutative geometry is a branch of mathematics concerned with geometric approach to noncommutative algebras, and with construction of spaces which are locally presented by noncommutative algebras of functions...
has become a particularly active discipline in ring theory.
First example: the integers
The most familiar example of a ring is the set of all
integerThe integers are formed by the natural numbers together with the negatives of the nonzero natural numbers .They are known as Positive and Negative Integers respectively...
s, Z, consisting of the
numberA number is a mathematical object used to count and measure. In mathematics, the definition of number has been extended over the years to include such numbers as zero, negative numbers, rational numbers, irrational numbers, and complex numbers....
s
 ..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ...
together with the usual operations of addition and multiplication. These operations satisfy the following properties:
 The integers form an abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under addition; that is:
 Closure axiom for addition
In mathematics, a set is said to be closed under some operation if performance of that operation on members of the set always produces a unique member of the same set. For example, the real numbers are closed under subtraction, but the natural numbers are not: 3 and 8 are both natural numbers, but...
: Given two integers a and b, their sum, a + b is also an integer.
 Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...
of addition: For any integers, a, b and c, (a + b) + c = a + (b + c). So, adding b to a, and then adding c to this result, is the same as adding c to b, and then adding this result to a.
 Existence of additive identity
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
: For any integer a, a + 0 = 0 + a = a. Zero is called the identity element of the integers because adding 0 to any integer (in any order) returns the same integer.
 Existence of additive inverse
In abstract algebra, the idea of an inverse element generalises the concept of a negation, in relation to addition, and a reciprocal, in relation to multiplication. The intuition is of an element that can 'undo' the effect of combination with another given element...
: For any integer a, there exists an integer denoted by −a such that a + (−a) = (−a) + a = 0. The element, −a, is called the additive inverse of a because adding a to −a (in any order) returns the identity.
 Commutativity
In mathematics an operation is commutative if changing the order of the operands does not change the end result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it...
of addition: For any two integers a and b, a + b = b + a. So the order in which two integers are added is irrelevant.
 The integers form a multiplicative monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
(a monoid under multiplication); that is:
 Closure axiom for multiplication: Given two integers a and b, their product, a · b is also an integer.
 Associativity of multiplication: Given any integers, a, b and c, (a · b) · c = a · (b · c). So multiplying b with a, and then multiplying c to this result, is the same as multiplying c with b, and then multiplying a to this result.
 Existence of multiplicative identity: For any integer a, a · 1 = 1 · a = a. So multiplying any integer with 1 (in any order) gives back that integer. One is therefore called the multiplicative identity.
 Multiplication is distributive
In mathematics, and in particular in abstract algebra, distributivity is a property of binary operations that generalizes the distributive law from elementary algebra.For example:...
over addition : These two structures on the integers (addition and multiplication) are compatible in the sense that
 a · (b + c) = (a · b) + (a · c), and
 (a + b) · c = (a · c) + (b · c)
 for any three integers a, b and c.
Formal definition
There are some differences in exactly what axioms are used to define a ring. Here one set of axioms for a ring with identity is given, and comments on variations follow.
A ring is a set R equipped with two
binary operationIn mathematics, a binary operation is a calculation involving two operands, in other words, an operation whose arity is two. Examples include the familiar arithmetic operations of addition, subtraction, multiplication and division....
s + : R × R → R and · : R × R → R (where × denotes the
Cartesian productIn mathematics, a Cartesian product is a construction to build a new set out of a number of given sets. Each member of the Cartesian product corresponds to the selection of one element each in every one of those sets...
), called addition and multiplication. To qualify as a ring, the set and two operations, (R, +, · ), must satisfy the following requirements known as the ring axioms.
 (R, +) is required to be an abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under addition:
1. 
Closure under addition. 
For all a, b in R, the result of the operation a + b is also in R. 
2. 
Associativity of addition. 
For all a, b, c in R, the equation (a + b) + c = a + (b + c) holds. 
3. 
Existence of additive identity. 
There exists an element 0 in R, such that for all elements a in R, the equation 0 + a = a + 0 = a holds. 
4. 
Existence of additive inverse. 
For each a in R, there exists an element b in R such that a + b = b + a = 0 
5. 
Commutativity of addition. 
For all a, b in R, the equation a + b = b + a holds. 
 (R, ·) is required to be a monoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
under multiplication:
1. 
Closure under multiplication. 
For all a, b in R, the result of the operation a · b is also in R. 
2. 
Associativity of multiplication. 
For all a, b, c in R, the equation (a · b) · c = a · (b · c) holds. 
3. 
Existence of multiplicative identity. 
There exists an element 1 in R, such that for all elements a in R, the equation 1 · a = a · 1 = a holds. 
1. For all a, b and c in R, the equation a · (b + c) = (a · b) + (a · c) holds. 
2. For all a, b and c in R, the equation (a + b) · c = (a · c) + (b · c) holds. 
This definition assumes that a binary operation on R is a
functionIn mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
defined on R×R with values in R. Therefore, for any a and b in R, the addition a + b and the product a · b are elements of R.
The most familiar example of a ring is the set of all
integerThe integers are formed by the natural numbers together with the negatives of the nonzero natural numbers .They are known as Positive and Negative Integers respectively...
s, Z = {..., −4, −3, −2, −1, 0, 1, 2, 3, 4, ... }, together with the usual operations of addition and multiplication.
Another familiar example is the set of
real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s R, equipped with the usual addition and multiplication.
Another example of a ring is the set of all square matrices of a fixed size, with real elements, using the matrix addition and multiplication of
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
. In this case, the ring elements 0 and 1 are the zero matrix (with all entries equal to 0) and the
identity matrixIn linear algebra, the identity matrix or unit matrix of size n is the n×n square matrix with ones on the main diagonal and zeros elsewhere. It is denoted by In, or simply by I if the size is immaterial or can be trivially determined by the context...
, respectively.
Second example: the ring Z_{4}
Consider the set Z
_{4} consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows (note that for any integer x, x
modIn mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
4 is defined to be the remainder when x is divided by 4):
 For any x, y in Z_{4}, x + y is defined to be their sum in Z (the set of all integers) mod 4
In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" after they reach a certain value—the modulus....
. So we can represent the additive structure of Z_{4} by the leftmost table as shown.
 For any x, y in Z_{4}, x ⋅ y is defined to be their product in Z (the set of all integers) mod 4. So we can represent the multiplicative structure of Z_{4} by the rightmost table as shown.
·  0  1  2  3 

0 
0 
0 
0 
0 
1 
0 
1 
2 
3 
2 
0 
2 
0 
2 
3 
0 
3 
2 
1 
+  0  1  2  3 
0 
0 
1 
2 
3 
1 
1 
2 
3 
0 
2 
2 
3 
0 
1 
3 
3 
0 
1 
2 
It is simple (but tedious) to verify that Z
_{4} is a ring under these operations. First of all, one can use the leftmost table to show that Z
_{4} is closed under addition (any result is either 0, 1, 2 or 3). Associativity of addition in Z
_{4} follows from associativity of addition in the set of all integers. The additive identity is 0 as can be verified by looking at the leftmost table. Given an integer x, there is always an inverse of x; this inverse is given by 4  x as one can verify from the additive table. Therefore, Z
_{4} is an
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under addition.
Similarly, Z
_{4} is closed under multiplication as the rightmost table shows (any result above is either 0, 1, 2 or 3). Associativity of multiplication in Z
_{4} follows from associativity of multiplication in the set of all integers. The multiplicative identity is 1 as can be verified by looking at the rightmost table. Therefore, Z
_{4} is a
monoidIn abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
under multiplication.
Distributivity of the two operations over each other follow from distributivity of addition over multiplication (and viceversa) in Z (the set of all integers).
Therefore, this set does indeed form a ring under the given operations of addition and multiplication.
Properties of this ring
 In general, given any two integers, x and y, if x ⋅ y = 0, then either x is 0 or y is 0. It is interesting to note that this does not hold for the ring (Z_{4}, +, ⋅):

 2 ⋅ 2 = 0
 although neither factor
In mathematics, factorization or factoring is the decomposition of an object into a product of other objects, or factors, which when multiplied together give the original...
is 0. In general, a nonzero element a of a ring, (R, +, ⋅) is said to be a zero divisorIn abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
in (R, +, ⋅), if there exists a nonzero element b of R such that a ⋅ b = 0. So in this ring, the only zero divisor is 2 (note that 0 ⋅ a = 0 for any a in a ring (R, +, ⋅) so 0 is not considered to be a zero divisor).
 A commutative ring which has no zero divisors is called an integral domain (see below). So Z, the ring of all integers (see above), is an integral domain (and therefore a ring), although Z_{4} (the above example) does not form an integral domain (but is still a ring). So in general, every integral domain is a ring but not every ring is an integral domain.
Third example: the trivial ring
If we define on the singleton set {0}:
0 + 0 = 0
0 × 0 = 0
then one can verify that ({0}, +, ×) forms a ring known as the
trivial ring. Since there can be only one result for any product or sum (0), this ring is both closed and associative for addition and multiplication, and furthermore satisfies the distributive law. The additive and multiplicative identities are both equal to 0. Similarly, the additive inverse of 0 is 0. The trivial ring is also a (rather trivial) example of a
zero ringIn ring theory, a branch of mathematics, a zero ring is a ring in which the product of any two elements is 0 . In ring theory, a branch of mathematics, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element). In ring theory, a branch of...
(see below).
Subring
Informally, a
subringIn mathematics, a subring of R is a subset of a ring, is itself a ring with the restrictions of the binary operations of addition and multiplication of R, and which contains the multiplicative identity of R...
of a ring is another ring that uses the "same" operations and is contained in it. More formally, suppose (R, +, ·) is a ring, and S is a subset of R such that
 for every a, b in S, a + b is in S;
 for every a, b in S, a · b is in S;
 for every a in S, the additive inverse −a of a is in S; and
 the multiplicative identity '1' of R is in S.
Let '+
_{S}' and '·
_{S}' denote the operations '+' and '·', restricted to S×S. Then (S, +
_{S}, ·
_{S}) is a subring of (R, +, ·). Since the restricted operations are completely determined by S and the original ones, the subring is often written simply as (S, +, ·).
For example, a subring of the complex number ring C is any subset of C that includes 1 and is closed under addition, multiplication, and negation, such as:
 The rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s Q
 The algebraic number
In mathematics, an algebraic number is a number that is a root of a nonzero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s A
 The real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s R
If A is a subring of R, and B is a
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
of A such that B is also a subring of R, then B is a subring of A.
Homomorphism
A
homomorphismIn ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
from a ring (R, +, ·) to a ring (S, ‡, *) is a function f from R to S that commutes with the ring operations; namely, such that, for all a, b in R the following identities hold:
Moreover, the function f must take the identity element 1
_{R} of '·' to the identity element 1
_{S} of '*'.
(this is not required if identity is not required).
For example, the function that maps each integer x to its remainder modulo 4 (a number in {0, 1, 2, 3}) is a homomorphism from the ring Z to the ring Z
_{4}.
A ring homomorphism is said to be an isomorphism if there exists an inverse homomorphism to f (i.e., a ring homomorphism which is an
inverse functionIn mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ=y, and g=x...
). Equivalently, any
bijectiveA bijection is a function giving an exact pairing of the elements of two sets. A bijection from the set X to the set Y has an inverse function from Y to X. If X and Y are finite sets, then the existence of a bijection means they have the same number of elements...
ring homomorphism is a ring isomorphism.
Ideal
The purpose of an
idealIn ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
in a ring is to somehow allow one to define the
quotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
of a ring (analogous to the
quotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of a
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
; see below). An ideal in a ring can therefore be thought of as a generalization of a
normal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
in a group. More formally, let (R, +, · ) be a ring. A
subsetIn mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
I of R is said to be a right ideal in R if:
 (I, +) is a subgroup of the underlying additive group in (R, +, · ) (i.e. (I, +) is a subgroup of (R, +)).
 For every x in I and r in R, x · r is in I.
A left ideal is similarly defined with the second condition being replaced. More specifically, a subset I of R is a left ideal in R if:
 (I, +) is a subgroup of the underlying additive group in (R, +, · ) (i.e. (I, +) is a subgroup of (R, +)).
 For every x in I and r in R, r · x is in I.
Notes
 If k is in R, then the set of elements k · R is a right ideal in R, and the set of elements R · k is a left ideal in R. These ideals (for any k in R) are called the principal
In ring theory, a branch of abstract algebra, a principal ideal is an ideal I in a ring R that is generated by a single element a of R.More specifically:...
right and left ideals generated by k.
 If every ideal in a ring (R, +, · ) is a principal ideal in (R, +, · ), (R, +, · ) is said to be a principal ideal ring
In mathematics, a principal right ideal ring is a ring R in which every right ideal is of the form xR for some element x of R...
.
 An ideal in a ring, (R, +, · ), is said to be a twosided ideal if it is both a left ideal and right ideal in (R, +, · ). It is preferred to call a twosided ideal, simply an ideal.
 If I = {0} (where 0 is the additive identity of the ring (R, +, · )), then I is an ideal known as the trivial ideal. Similarly, R is also an ideal in (R, +, · ) called the unit ideal.
Examples
 Any additive subgroup of the integers is an ideal in the integers with its natural ring structure.
 There are no nontrivial ideals in R (the ring of all real numbers) (i.e., the only ideals in R are {0} and R itself). More generally, a field
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
cannot contain any nontrivial ideals.
 From the previous example, every field must be a principal ideal ring.
 A subset, I, of a commutative ring (R, +, · ) is a left ideal if and only if
IFF, Iff or iff may refer to:Technology/Science:* Identification friend or foe, an electronic radiobased identification system using transponders...
it is a right ideal. So for simplicity's sake, we refer to any ideal in a commutative ring as just an ideal.
History
The study of rings originated from the theory of
polynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
s and the theory of
algebraic integerIn number theory, an algebraic integer is a complex number that is a root of some monic polynomial with coefficients in . The set of all algebraic integers is closed under addition and multiplication and therefore is a subring of complex numbers denoted by A...
s. Furthermore, the appearance of
hypercomplex numberIn mathematics, a hypercomplex number is a traditional term for an element of an algebra over a field where the field is the real numbers or the complex numbers. In the nineteenth century number systems called quaternions, tessarines, coquaternions, biquaternions, and octonions became established...
s in the mid19th century undercut the preeminence of
fieldsIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
in mathematical analysis.
In the 1880s Richard Dedekind introduced the concept of a ring, and the term ring (Zahlring) was coined by
David HilbertDavid Hilbert was a German mathematician. He is recognized as one of the most influential and universal mathematicians of the 19th and early 20th centuries. Hilbert discovered and developed a broad range of fundamental ideas in many areas, including invariant theory and the axiomatization of...
in 1892 and published in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897. According to Harvey Cohn, Hilbert used the term for a specific ring that had the property of "circling directly back" to an element of itself.
The first axiomatic definition of a ring was given by Adolf Fraenkel in an essay in Journal für die reine und angewandte Mathematik (A. L. Crelle), vol. 145, 1914. In 1921,
Emmy NoetherAmalie Emmy Noether was an influential German mathematician known for her groundbreaking contributions to abstract algebra and theoretical physics. Described by David Hilbert, Albert Einstein and others as the most important woman in the history of mathematics, she revolutionized the theories of...
gave the first axiomatic foundation of the theory of
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s in her monumental paper Ideal Theory in Rings.
Elementary properties of rings
Some properties of rings follow directly from the ring axioms through
very simple proofsThe following proofs of elementary ring properties use only the axioms that define a mathematical ring:Multiplication by zero:Theorem: 0 ⋅ a = a ⋅ 0 = 0Trivial ring:...
.
In particular, since the axioms state that (R,+) is a commutative group, all pertinent theorems of
group theoryIn mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other wellknown algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
apply, such as the uniqueness of the additive identity and the uniqueness of the additive inverse of a particular element. In the same way one can prove the uniqueness of the inverse of a
unit in a ringIn mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
.
However, rings also have specific properties that combine addition with multiplication. In any ring (R, +, ⋅):
 For any element a, the equations a ⋅ 0 = 0 ⋅ a = 0 hold.
 If 0 = 1, the ring is trivial (that is, R has only one element).
 For any element a, the equation (−1) ⋅ a = −a holds.
 For any elements a and b, the equations (−a) ⋅ b = a ⋅ (−b) = −(a ⋅ b) hold.
Quotient ring
Informally, the
quotient ringIn ring theory, a branch of modern algebra, a quotient ring, also known as factor ring or residue class ring, is a construction quite similar to the factor groups of group theory and the quotient spaces of linear algebra...
of a ring, is a generalization of the notion of a
quotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
of a group. More formally, given a ring (R, +, · ) and a
twosided idealIn ring theory, a branch of abstract algebra, an ideal is a special subset of a ring. The ideal concept allows the generalization in an appropriate way of some important properties of integers like "even number" or "multiple of 3"....
I of (R, +, · ), the quotient ring (or factor ring) R/I is the set of cosets of I (with respect to the underlying additive group of (R, +, · ); i.e. cosets with respect to (R, +)) together with the operations:
+ (b + I) = (a + b) + I and(b + I) = (ab) + I.
For every a, b in R.
Polynomial ring
Formal definition
Let (R, +
_{R}, ·
_{R}) be a ring and let
where the convention that
is the set of all nonnegative integers, is adopted. Define +
_{S} : S × S → S and ·
_{S} : S × S → S, as follows (where
and
are arbitrary elements of S):
Then (S, +
_{S}, ·
_{S}) is a ring referred to as the polynomial ring over R.
Notes
Most authors write S as R[X], where
is an element of S. This convention is adopted because if
is an element of S, and if
, f can be written as a polynomial with coefficients in R, in the variable X, as follows:
This allows one to view S as merely the set of all polynomials over R in the variable X, with multiplication and addition of polynomials defined in the canonical manner. Therefore, in all that follows, S, denoted by R[X], shall be identified in this fashion. Addition and multiplication in S, and those of the underlying ring R, will be denoted by juxtaposition.
Matrix ring
Formal definition
Let (R, +
_{R}, ·
_{R}) be a ring and let
. Define +
_{M} : M
_{r}(R) × M
_{r}(R) → M
_{r}(R) and ·
_{M} : M
_{r}(R) × M
_{r}(R) → M
_{r}(R), as follows (where
and
are arbitrary elements of M
_{r}(R)):
Then (M
_{r}(R), +
_{M}, ·
_{M}) is a ring referred to as the ring of r×r matrices over R.
Notes
The ring M
_{r}(R) can be viewed as merely the ring of r×r matrices over R with matrix addition and multiplication defined in the canonical manner. For given
in M
_{r}(R), f can be identified with the r×r matrix whose (i, j)entry is
. Therefore, in all that follows, M
_{r}(R) and each of its elements shall be identified in this fashion. Addition and multiplication in M
_{r}(R), and those of the underlying ring R, will be denoted by juxtaposition.
Some examples of the ubiquity of rings
Many different kinds of
mathematical objectIn mathematics and the philosophy of mathematics, a mathematical object is an abstract object arising in mathematics.Commonly encountered mathematical objects include numbers, permutations, partitions, matrices, sets, functions, and relations...
s can be fruitfully analyzed in terms of some
associated ringIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
.
Cohomology ring of a topological space
To any
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
X one can associate its integral
cohomology ringIn mathematics, specifically algebraic topology, the cohomology ring of a topological space X is a ring formed from the cohomology groups of X together with the cup product serving as the ring multiplication. Here 'cohomology' is usually understood as singular cohomology, but the ring structure is...
a graded ring. There are also homology groups
of a space, and indeed these were defined first, as a useful tool for distinguishing between certain pairs of topological spaces, like the
sphereA sphere is a perfectly round geometrical object in threedimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
s and
toriIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
, for which the methods of pointset topology are not wellsuited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the
universal coefficient theoremIn mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A...
. However, the advantage of the cohomology groups is that there is a
natural productIn mathematics, specifically in algebraic topology, the cup product is a method of adjoining two cocycles of degree p and q to form a composite cocycle of degree p + q. This defines an associative graded commutative product operation in cohomology, turning the cohomology of a space X into a...
, which is analogous to the observation that one can multiply pointwise a k
multilinear form and an lmultilinear form to get a (k + l)mulilinear form.
The ring structure in cohomology provides the foundation for
characteristic classIn mathematics, a characteristic class is a way of associating to each principal bundle on a topological space X a cohomology class of X. The cohomology class measures the extent to which the bundle is "twisted" — particularly, whether it possesses sections or not...
es of
fiber bundleIn mathematics, and particularly topology, a fiber bundle is intuitively a space which locally "looks" like a certain product space, but globally may have a different topological structure...
s, intersection theory on manifolds and
algebraic varietiesIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
,
Schubert calculusIn mathematics, Schubert calculus is a branch of algebraic geometry introduced in the nineteenth century by Hermann Schubert, in order to solve various counting problems of projective geometry...
and much more.
Burnside ring of a group
To any
groupIn mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is associated its
Burnside ringIn mathematics, the Burnside ring of a finite group is an algebraic construction that encodes the different ways the group can act on finite sets...
which uses a ring to describe the various ways the group can
actIn algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
on a finite set. The Burnside ring's additive group is the
free abelian groupIn abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the
representation ringIn mathematics, especially in the area of algebra known as representation theory, the representation ring of a group is a ring formed from all the linear representations of the group. For a given group, the ring will depend on the base field of the representations...
: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module. The ring structure allows a formal way of subtracting one action from another. Since the Burnside ring is contained as a finite index subring of the representation ring, one can pass easily from one to the other by extending the coefficients from integers to the rational numbers.
Representation ring of a group ring
To any
group ringIn algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring and its basis is onetoone with the given group. As a ring, its addition law is that of the free...
or
Hopf algebraIn mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antiautomorphism satisfying a certain property.Hopf algebras occur naturally...
is associated its
representation ringIn mathematics, especially in the area of algebra known as representation theory, the representation ring of a group is a ring formed from all the linear representations of the group. For a given group, the ring will depend on the base field of the representations...
or "Green ring". The representation ring's additive group is the free abelian group whose basis are the indecomposable modules and whose addition corresponds to the direct sum. Expressing a module in terms of the basis is finding an indecomposable decomposition of the module. The multiplication is the tensor product. When the algebra is semisimple, the representation ring is just the character ring from
character theoryIn mathematics, more specifically in group theory, the character of a group representation is a function on the group which associates to each group element the trace of the corresponding matrix....
, which is more or less the
Grothendieck groupIn mathematics, the Grothendieck group construction in abstract algebra constructs an abelian group from a commutative monoid in the best possible way...
given a ring structure.
Function field of an irreducible algebraic variety
To any irreducible
algebraic varietyIn mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
is associated its
function fieldIn algebraic geometry, the function field of an algebraic variety V consists of objects which are interpreted as rational functions on V...
. The points of an algebraic variety correspond to
valuation ringIn abstract algebra, a valuation ring is an integral domain D such that for every element x of its field of fractions F, at least one of x or x −1 belongs to D....
s contained in the function field and containing the coordinate ring. The study of
algebraic geometryAlgebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
makes heavy use of
commutative algebraCommutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
to study geometric concepts in terms of ringtheoretic properties.
Birational geometryIn mathematics, birational geometry is a part of the subject of algebraic geometry, that deals with the geometry of an algebraic variety that is dependent only on its function field. In the case of dimension two, the birational geometry of algebraic surfaces was largely worked out by the Italian...
studies maps between the subrings of the function field.
Face ring of a simplicial complex
Every
simplicial complexIn mathematics, a simplicial complex is a topological space of a certain kind, constructed by "gluing together" points, line segments, triangles, and their ndimensional counterparts...
has an associated face ring, also called its Stanley–Reisner ring. This ring reflects many of the combinatorial properties of the simplicial complex, so it is of particular interest in
algebraic combinatoricsAlgebraic combinatorics is an area of mathematics that employs methods of abstract algebra, notably group theory and representation theory, in various combinatorial contexts and, conversely, applies combinatorial techniques to problems in algebra....
. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of
simplicial polytopeIn geometry, a simplicial polytope is a dpolytope whose facets are all simplices.For example, a simplicial polyhedron contains only triangular faces and corresponds via Steinitz's theorem to a maximal planar graph....
s.
Finite ring
Given a natural number m, how many distinct rings (not necessarily with unity) have m elements? When m is prime there are only two rings of order m; the additive group of each being isomorphic to the cyclic group of order m. One is the
zero ringIn ring theory, a branch of mathematics, a zero ring is a ring in which the product of any two elements is 0 . In ring theory, a branch of mathematics, a zero ring is a ring (without unity) in which the product of any two elements is 0 (the additive identity element). In ring theory, a branch of...
and the other is the Galois field.
As with
finite groupIn mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
s, the complexity of the classification depends upon the complexity of the prime factorization of m. If m is the square of a prime, for instance, there are precisely eleven rings having order m. On the other hand, there can be only two groups having order m; both of which are abelian.
The theory of finite rings is more complex than that of finite abelian groups, since any finite abelian group is the additive group of at least two nonisomorphic finite rings: the direct product of copies of
, and the zero ring. On the other hand, the theory of finite rings is simpler than that of not necessarily abelian finite groups. For instance, the
classification of finite simple groupsIn mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four categories described below. These groups can be seen as the basic building blocks of all finite groups, in much the same way as the prime numbers are the basic...
was one of the major breakthroughs of 20th century mathematics, its proof spanning thousands of journal pages. On the other hand, any finite
simple ringIn abstract algebra, a simple ring is a nonzero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
is isomorphic to the ring
of n by n matrices over a finite field of order q. This follows from two theorems of
Joseph WedderburnJoseph Henry Maclagan Wedderburn was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, and part of the Artin–Wedderburn theorem on simple algebras...
established in 1905 and 1907.
One of these theorems, known as
Wedderburn's little theoremIn mathematics, Wedderburn's little theorem states that every finite domain is a field. In other words, for finite rings, there is no distinction between domains, skewfields and fields.The Artin–Zorn theorem generalizes the theorem to alternative rings....
, asserts that any finite
division ringIn abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with...
is necessarily commutative.
Nathan JacobsonNathan Jacobson was an American mathematician....
later discovered yet another condition which guarantees commutativity of a ring:
If for every element r of R there exists an integer such that , then R is commutative.
If, r
^{2} = r for every r, the ring is called a
Boolean ringIn mathematics, a Boolean ring R is a ring for which x2 = x for all x in R; that is, R consists only of idempotent elements....
. More general conditions which guarantee commutativity of a ring are also known.
The number of rings with m elements, for m a natural number, is listed under
A027623 in the
OnLine Encyclopedia of Integer SequencesThe OnLine Encyclopedia of Integer Sequences , also cited simply as Sloane's, is an online database of integer sequences, created and maintained by N. J. A. Sloane, a researcher at AT&T Labs...
.
Associative algebras
An
associative algebraIn mathematics, an associative algebra A is an associative ring that has a compatible structure of a vector space over a certain field K or, more generally, of a module over a commutative ring R...
is a ring that is also a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
over a field K. For instance, the set of n by n matrices over the real field R has dimension n
^{2} as a real vector space, and the
matrix multiplicationIn mathematics, matrix multiplication is a binary operation that takes a pair of matrices, and produces another matrix. If A is an nbym matrix and B is an mbyp matrix, the result AB of their multiplication is an nbyp matrix defined only if the number of columns m of the left matrix A is the...
corresponds to the ring multiplication. For a nontrivial but elementary example consider 2 × 2 real matrices.
Lie ring
A
Lie ringIn mathematics a Lie ring is a structure related to Lie algebras that can arise as a generalisation of Lie algebras, or through the study of the lower central series of groups. Formal definition :...
is defined to be a ring that is
nonassociativeIn abstract algebra, a nonassociative ring is a generalization of the concept of ring.A nonassociative ring is a set R with two operations, addition and multiplication, such that:# R is an abelian group under addition:## a+b = b+a...
and anticommutative under multiplication, that also satisfies the
Jacobi identityIn mathematics the Jacobi identity is a property that a binary operation can satisfy which determines how the order of evaluation behaves for the given operation. Unlike for associative operations, order of evaluation is significant for operations satisfying Jacobi identity...
. More specifically we can define a Lie ring
to be an
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under addition with an operation
that has the following properties:

 for all x, y, z ∈ L.


 for all x, y, z in L.

Lie rings need not be
Lie groupIn mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...
s under addition. Any
Lie algebraIn mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. Lie algebras were introduced to study the concept of infinitesimal transformations. The term "Lie algebra" was introduced by Hermann Weyl in the...
is an example of a Lie ring. Any associative ring can be made into a Lie ring by defining a bracket operator
. Conversely to any Lie algebra there is a corresponding ring, called the
universal enveloping algebraIn mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the nonassociative structure L to a unital associative algebra which captures the important properties of L.Any associative algebra A over the field K becomes a Lie algebra...
.
Lie rings are used in the study of finite
pgroupIn mathematics, given a prime number p, a pgroup is a periodic group in which each element has a power of p as its order: each element is of prime power order. That is, for each element g of the group, there exists a nonnegative integer n such that g to the power pn is equal to the identity element...
s through the Lazard correspondence. The lower central factors of a pgroup are finite abelian pgroups, so modules over Z/pZ. The direct sum of the lower central factors is given the structure of a Lie ring by defining the bracket to be the
commutatorIn mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory.Group theory:...
of two coset representatives. The Lie ring structure is enriched with another module homomorphism, then pth power map, making the associated Lie ring a socalled restricted Lie ring.
Lie rings are also useful in the definition of a padic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the padic integers. The definition of finite groups of Lie type due to Chevalley involves restricting from a Lie algebra over the complex numbers to a Lie algebra over the integers, and the reducing modulo p to get a Lie algebra over a finite field.
Topological ring
Let (X, T) is a
topological spaceTopological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
and (X, +, · ) be a ring. Then (X, T, +, · ) is said to be a
topological ringIn mathematics, a topological ring is a ring R which is also a topological space such that both the addition and the multiplication are continuous as mapswhere R × R carries the product topology. General comments :...
, if its ring structure and topological structure are both compatible (i.e work together) over each other. That is, the addition map (
) and the multiplication map (
) have to be both continuous as maps between topological spaces where X x X inherits the
product topologyIn topology and related areas of mathematics, a product space is the cartesian product of a family of topological spaces equipped with a natural topology called the product topology...
. So clearly, any topological ring is a
topological groupIn mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a...
(under addition).
Examples
 The set of all real numbers, R, with its natural ring structure and the standard topology forms a topological ring.
 The direct product of two topological rings is also a topological ring.
Commutative rings
Although ring addition is commutative, so that for every a, b in R, a + b = b + a, ring multiplication is not required to be commutative; a · b need not equal b · a for all a, b in R. Rings that also satisfy commutativity for multiplication are called
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s. Formally,
Formal definition
Let (R, +, · ) be a ring. Then (R, +, · ) is said to be a commutative ring if for every a, b in R, a · b = b · a. That is, (R, +, · ) is required to be a commutative monoid under multiplication.
Examples
 The integers form a commutative ring under the natural operations of addition and multiplication.
 An example of a noncommutative ring is the ring of n × n matrices
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
over a nontrivial fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
K, for n > 1. In particular, the ring of all 2 × 2 matrices over R (the set of all real numberIn mathematics, a real number is a value that represents a quantity along a continuum, such as 5 , 4/3 , 8.6 , √2 and π...
s) do not form a commutative ring as the following computationComputation is defined as any type of calculation. Also defined as use of computer technology in Information processing.Computation is a process following a welldefined model understood and expressed in an algorithm, protocol, network topology, etc...
shows:
, which is not equal to
Principal ideal domains
Although rings are structurally similar to the integers, there are certain ringtheoretic properties that the integers may satisfy but a general ring may not. One such property is the requirement that every ideal in a ring be generated by a single element; that is, be a principal ideal. Formally,
Definition
Let R be a ring. Then R is said to be a principal ideal ring (abbreviated PIR), if every ideal in R is of the form a · R = {a · r  r in R}. A principal ideal domain (PID) is a principal ideal ring that is also an integral domain.
The requirement that a ring be a principal ideal domain is somewhat stronger than the other more common properties a ring may satisfy. For example, it is true that if a ring, R, is a
unique factorization domainIn mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
(UFD), then the
polynomial ringIn mathematics, especially in the field of abstract algebra, a polynomial ring is a ring formed from the set of polynomials in one or more variables with coefficients in another ring. Polynomial rings have influenced much of mathematics, from the Hilbert basis theorem, to the construction of...
over R is also a UFD. However, such a result does not in general hold for principal ideal rings. For example, the integers are an easy example of a principal ideal ring, but the polynomial ring over the integers fails to be a PIR; if R = Z[x] denotes the polynomial ring over the integers, I = 2 · R + X · R is an ideal which cannot be generated by a single element. Despite this counterexample, the polynomial ring over any field is always a principal ideal domain and in fact, a
Euclidean domainIn mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
. More generally, a polynomial ring is a PID if and only if the polynomial ring in question is over a field.
Aside from the polynomial ring over a PIR, principal ideal rings possess many interesting properties due to their connection with the integers in terms of divisibility; that is, principal ideal domains behave similarly to the integers with respect to divisibility. For example, any PID is a UFD; i.e., an analogue of the
fundamental theorem of arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
holds for principal ideal domains. Furthermore, since Noetherian rings are precisely those rings in which any ideal is finitely generated, principal ideal domains are trivially Noetherian rings. The fact that irreducible elements coincide with prime elements for PID's, together with the fact that every PID is Noetherian, implies that any PID is a UFD. One can also speak of the greatest common divisor of two elements in a PID; if x and y are elements of R, a principal ideal domain, then x · R + y · R = c · R for some c in R since the lefthand side is indeed an ideal. Therefore, c is the desired "GCD" of x and y.
An important class of rings lying in between fields and PID's, is the class of Euclidean domains. In particular, any field is a Euclidean domain, and any Euclidean domain is a PID. An ideal in a Euclidean domain is generated by any element of that ideal with minimum degree (all such elements must be associate). However, not every PID is a Euclidean domain; the ring
furnishes a counterexample.
Unique factorization domains
The theory of unique factorization domains (UFD) also forms an important part of ring theory. In effect, a unique factorization domain is ring in which an analogue of the
fundamental theorem of arithmeticIn number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
holds. Formally,
Definition
Let R be a ring. Then R is said to be a unique factorization domain (abbreviated UFD), if the following conditions are satisfied:
1. R is an integral domain.
2. Every nonzero
nonunitIn mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
of R is the product of a finite number of irreducible elements.
3. If
=
, where all a
_{i}'s and b
_{j}'s are irreducible, then n = m and after possible renumbering of the a
_{i}'s and b
_{j}'s, b
_{i} = a
_{i} · u
_{i} where u
_{i} is a
unitIn mathematics, an invertible element or a unit in a ring R refers to any element u that has an inverse element in the multiplicative monoid of R, i.e. such element v that...
in R.
The second condition above guarantees that "nontrivial" elements of R can be decomposed into irreducibles, and according to the third condition, such a finite decomposition is unique "up to multiplication by unit elements." This weakened form of uniqueness is reasonable to assume for otherwise even the integers would not satisfy the properties of being a UFD ((2)
^{2} = 2
^{2} = 4 demonstrates two "distinct" decompositions of 4; however both decompositions of 4 are equivalent up to multiplication by units (1 and +1)). The fact that the integers constitute a UFD follows from the fundamental theorem of arithmetic.
For arbitrary rings, one may define a
prime elementIn abstract algebra, an element p of a commutative ring R is said to be prime if it is not zero, not a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b...
and an
irreducible elementIn abstract algebra, a nonzero nonunit element in an integral domain is said to be irreducible if it is not a product of two nonunits.Irreducible elements should not be confused with prime elements...
; these two may not in general coincide. However, a prime element in a domain is always irreducible. For UFD's, irreducible elements are also primes.
The class of unique factorization domains is related to other classes of rings. For instance, any
Noetherian domainIn mathematics, more specifically in the area of modern algebra known as ring theory, a Noetherian ring, named after Emmy Noether, is a ring in which every nonempty set of ideals has a maximal element...
satisfies conditions 1 and 2 above, but in general Noetherian domains fail to satisfy condition 3. However, if the set of prime elements and the set of irreducible elements coincide for a Noetherian domain, the third condition of a UFD is satisfied. In particular, principal ideal domains are UFD's.
Integral domains and fields
While rings are very important mathematical objects, there are a lot of restrictions involved in their theory. For instance, suppose we have a ring R and suppose a and b are in R. If a is nonzero and a · b = 0, then b need not necessarily be 0. In particular, if a · b = a · c with a nonzero b need not equal c. An example of this is the set of n x n matrices over R where a maybe such a nonzero matrix but may be singular. In this case, the result may not be true. However, we can impose additional conditions on the ring to ensure that this be true; namely make the ring into an
integral domain (that is a nontrivial
commutative ringIn ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
with no
zero divisorIn abstract algebra, a nonzero element a of a ring is a left zero divisor if there exists a nonzero b such that ab = 0. Similarly, a nonzero element a of a ring is a right zero divisor if there exists a nonzero c such that ca = 0. An element that is both a left and a right zero divisor is simply...
s). But we still run into a problem; namely we can't necessarily divide by nonzero elements. For example, the collection of all integers form an integral domain but we still can't divide an integer a by another integer b. For example, 2 cannot divide 3 to obtain another element in this ring. However, this problem can readily be solved if we ensure that every element in the ring has a multiplicative inverse. A
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
is a ring in which the nonzero elements form an
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
under multiplication. In particular, a field is an integral domain (and therefore has no zero divisors) along with an additional operation of 'division'. Namely, if a and b are in a field F, then a/b is defined to be a · b
^{−1} which is well defined.
Formal definition
Let (R, +, · ) be a ring. Then (R, +, · ) is said to be an integral domain if (R, +, · ) is commutative and has no zero divisors. Furthermore, (R, +, · ) is said to be a
fieldIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
, if its nonzero elements form an abelian group under multiplication.
Note
 If we require that the zero element (0) in a ring to have a multiplicative inverse, then the ring must be trivial.
Examples
 The integers form a commutative ring under the natural operations of addition and multiplication. In fact, the integers form what is known as an integral domain (a commutative ring with no zero divisors).
 A ring whose nonzero elements form an abelian group under multiplication (not just a commutative monoid), is called a field. So every field is an integral domain and every integral domain is a commutative ring. Furthermore, any finite integral domain is a field.
Relation to other algebraic structures
The following is a chain of class inclusions that describes the relationship between rings, domains and fields:
 Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
s ⊃ integral domains ⊃ half factorization domains ⊃unique factorization domainIn mathematics, a unique factorization domain is, roughly speaking, a commutative ring in which every element, with special exceptions, can be uniquely written as a product of prime elements , analogous to the fundamental theorem of arithmetic for the integers...
s ⊃ principal ideal domainIn abstract algebra, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are principal, although some authors refer to PIDs as...
s ⊃ Euclidean domainIn mathematics, more specifically in abstract algebra and ring theory, a Euclidean domain is a ring that can be endowed with a certain structure – namely a Euclidean function, to be described in detail below – which allows a suitable generalization of the Euclidean algorithm...
s ⊃ fieldsIn abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
Fields and integral domains are very important in modern algebra.
Noncommutative rings
The study of
noncommutative ringIn mathematics, more specifically modern algebra and ring theory, a noncommutative ring is a ring whose multiplication is not commutative; that is, if R is a noncommutative ring, there exists a and b in R with a·b ≠ b·a, and conversely.Noncommutative rings are ubiquitous in mathematics, and occur...
s is a major area in modern algebra; especially
ring theoryIn abstract algebra, ring theory is the study of rings—algebraic structures in which addition and multiplication are defined and have similar properties to those familiar from the integers...
. Often noncommutative rings possess interesting invariants that commutative rings do not. As an example, there exist rings which contain nontrivial proper left or right ideals, but are still
simpleIn abstract algebra, a simple ring is a nonzero ring that has no ideal besides the zero ideal and itself. A simple ring can always be considered as a simple algebra. This notion must not be confused with the related one of a ring being simple as a left module over itself...
; that is contain no nontrivial proper (twosided) ideals. This example illustrates how one must take care when studying noncommutative rings because of possible counterintuitive misconceptions.
The theory of
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
s is one illustration of a special case of an object studied in noncommutative ring theory. In
linear algebraLinear algebra is a branch of mathematics that studies vector spaces, also called linear spaces, along with linear functions that input one vector and output another. Such functions are called linear maps and can be represented by matrices if a basis is given. Thus matrix theory is often...
, the "scalars of a vector space" are required to lie in a field  a commutative
division ringIn abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a nontrivial ring in which every nonzero element a has a multiplicative inverse, i.e., an element x with...
. The concept of a
moduleIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, however, requires only that the scalars lie in an abstract ring. Neither commutativity nor the division ring assumption is required on the scalars in this case. Module theory has various applications in noncommutative ring theory, as one can often obtain information about the structure of a ring by making use of its modules. The concept of the
Jacobson radicalIn mathematics, more specifically ring theory, a branch of abstract algebra, the Jacobson radical of a ring R is an ideal which consists of those elements in R which annihilate all simple right Rmodules. It happens that substituting "left" in place of "right" in the definition yields the same...
of a ring; that is, the intersection of all right/left
annihilatorsIn mathematics, specifically module theory, annihilators are a concept that generalizes torsion and orthogonal complement.Definitions:Let R be a ring, and let M be a left Rmodule. Choose a nonempty subset S of M...
of
simpleIn mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no nonzero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a nonzero element of M equals M...
right/left modules over a ring, is one example. The fact that the Jacobson radical can be viewed as the intersection of all maximal right/left ideals in the ring, shows how the internal structure of the ring is reflected by its modules. It is also remarkable that the intersection of all maximal right ideals in a ring is the same as the intersection of all maximal left ideals in the ring, in the context of all rings; whether commutative or noncommutative. Therefore, the Jacobson radical also captures a concept which may seem to be not welldefined for noncommutative rings.
Noncommutative rings serve as an active area of research due to their ubiquity in mathematics. For instance, the ring of n by n
matrices over a fieldIn mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
is noncommutative despite its natural occurrence in
physicsPhysics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
. More generally,
endomorphism ringIn abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
s of abelian groups are rarely commutative.
Noncommutative rings, like noncommutative groups, are not very well understood. For instance, although every finite
abelian groupIn abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
is the
direct sum of (finite)
cyclic groupIn group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .Definition:A group G is called cyclic if there exists an element g...
s of primepower order, nonabelian groups do not possess such a simple structure. Likewise, various invariants exist for commutative rings, whereas invariants of noncommutative rings are difficult to find. As an example, the nilradical, although "innocent" in nature, need not be an ideal unless the ring is assumed to be commutative. Specifically, the set of all nilpotent elements in the ring of all n x n matrices over a division ring never forms an ideal, irrespective of the division ring chosen. Therefore, the nilradical cannot be studied in noncommutative ring theory. Note however that there are analogues of the nilradical defined for noncommutative rings, that coincide with the nilradical when commutativity is assumed.
One of the best known noncommutative rings is the division ring of quaternions.
Category theoretical description
Every ring can be thought of as a
monoidIn category theory, a monoid in a monoidal category is an object M together with two morphisms* \mu : M\otimes M\to M called multiplication,* and \eta : I\to M called unit,...
in Ab, the
category of abelian groupsIn mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category....
(thought of as a
monoidal categoryIn mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...
under the tensor product of
modules). The monoid action of a ring R on a abelian group is simply an
RmoduleIn abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
. Essentially, an Rmodule is a generalization of the notion of a
vector spaceA vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
 where rather than a vector space over a field, one has a "vector space over a ring".
Let (A, +) be an abelian group and let End(A) be its
endomorphism ringIn abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End...
(see above). Note that, essentially, End(A) is the set of all morphisms of A, where if f is in End(A), and g is in End(A), the following rules may be used to compute f + g and f · g:
where + as in f(x) + g(x) is addition in A, and function composition is denoted from right to left. Therefore,
associatedIn category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
to any abelian group, is a ring. Conversely, given any ring, (R, +, · ), (R, +) is an abelian group. Furthermore, for every r in R, right (or left) multiplication by r gives rise to a morphism of (R, +), by right (or left) distributivity. Let A = (R, +). Consider those
endomorphismIn mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...
s of A, that "factor through" right (or left) multiplication of R. In other words, let End
_{R}(A) be the set of all morphisms m of A, having the property that m(r · x) = r · m(x). It was seen that every r in R gives rise to a morphism of A  right multiplication by r. It is in fact true that this association of any element of R, to a morphism of A, as a function from R to End
_{R}(A), is an isomorphism of rings. In this sense, therefore, any ring can be viewed as the endomorphism ring of some abelian Xgroup (by Xgroup, it is meant a group with X being its
set of operatorsIn abstract algebra, a branch of pure mathematics, the algebraic structure group with operators or Ωgroup is a group with a set of group endomorphisms.Groups with operators were extensively studied by Emmy Noether and her school in the 1920s...
). In essence, the most general form of a ring, is the endomorphism group of some abelian Xgroup.
General references
 Atiyah M. F.
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
, Macdonald, I. G.Ian Grant Macdonald is a British mathematician known for his contributions to symmetric functions, special functions, Lie algebra theory and other aspects of algebraic combinatorics ....
, Introduction to commutative algebra. AddisonWesley Publishing Co., Reading, Mass.LondonDon Mills, Ont. 1969 ix+128 pp.
 Beachy, J. A. Introductory Lectures on Rings and Modules. Cambridge, England: Cambridge University Press, 1999.
 Dresden, G. "Small Rings." http://home.wlu.edu/~dresdeng/smallrings/
 Ellis, G. Rings and Fields. Oxford, England: Oxford University Press, 1993.
 Goodearl, K. R., Warfield, R. B., Jr., An introduction to noncommutative Noetherian rings. London Mathematical Society Student Texts, 16. Cambridge University Press, Cambridge, 1989. xviii+303 pp. ISBN 0521360862
 Herstein, I. N., Noncommutative rings. Reprint of the 1968 original. With an afterword by Lance W. Small. Carus Mathematical Monographs, 15. Mathematical Association of America, Washington, DC, 1994. xii+202 pp. ISBN 088385015X
 Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19–21, 1951
 Nathan Jacobson
Nathan Jacobson was an American mathematician....
, Structure of rings. American Mathematical Society Colloquium Publications, Vol. 37. Revised edition American Mathematical Society, Providence, R.I. 1964 ix+299 pp.
 Nathan Jacobson, The Theory of Rings. American Mathematical Society Mathematical Surveys, vol. I. American Mathematical Society, New York, 1943. vi+150 pp.
 Lam, T. Y., A first course in noncommutative rings. Second edition. Graduate Texts in Mathematics, 131. SpringerVerlag, New York, 2001. xx+385 pp. ISBN 0387951830
 Lam, T. Y., Exercises in classical ring theory. Second edition. Problem Books in Mathematics. SpringerVerlag, New York, 2003. xx+359 pp. ISBN 0387005005
 Lam, T. Y., Lectures on modules and rings. Graduate Texts in Mathematics, 189. SpringerVerlag, New York, 1999. xxiv+557 pp. ISBN 0387984283.
 McConnell, J. C.; Robson, J. C. Noncommutative Noetherian rings. Revised edition. Graduate Studies in Mathematics, 30. American Mathematical Society, Providence, RI, 2001. xx+636 pp. ISBN 0821821695
 Rowen, Louis H., Ring theory. Vol. I, II. Pure and Applied Mathematics, 127, 128. Academic Press, Inc., Boston, MA, 1988. ISBN 0125998414, ISBN 0125998422
 Sloane, N. J. A. Sequences A027623 and A037234 in "The OnLine Encyclopedia of Integer Sequences
 Zwillinger, D. (Ed.). "Rings." §2.6.3 in CRC Standard Mathematical Tables and Formulae. Boca Raton, FL: CRC Press, pp. 141–143, 1995
Special references
 Ballieu, R. "Anneaux finis; systèmes hypercomplexes de rang trois sur un corps commutatif." Ann. Soc. Sci. Bruxelles. Sér. I 61, 222227, 1947.
 Berrick, A. J. and Keating, M. E. An Introduction to Rings and Modules with KTheory in View. Cambridge, England: Cambridge University Press, 2000.
 Fine, B. "Classification of Finite Rings of Order ." Math. Mag. 66, 248252, 1993
 Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64, 922, 1980
 Fraenkel, A. "Über die Teiler der Null und die Zerlegung von Ringen." J. reine angew. Math. 145, 139176, 1914
 Gilmer, R. and Mott, J. "Associative Rings of Order ." Proc. Japan Acad. 49, 795799, 1973
 Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: SpringerVerlag, 1998
 Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: AddisonWesley, 1998
 Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: Dover, 2000
 Pierce, Richard S., Associative algebras. Graduate Texts in Mathematics, 88. Studies in the History of Modern Science, 9. SpringerVerlag, New York–Berlin, 1982. xii+436 pp. ISBN 0387906932
Historical references
 History of ring theory at the MacTutor Archive
 Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillian, 1996
 Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed. New York: SpringerVerlag, 2004. ISBN 3540434917
 Faith, Carl, Rings and things and a fine array of twentieth century associative algebra. Mathematical Surveys and Monographs, 65. American Mathematical Society, Providence, RI, 1999. xxxiv+422 pp. ISBN 0821809938
 Itô, K. (Ed.). "Rings." §368 in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 2. Cambridge, MA: MIT Press, 1986
 Kleiner, I. "The Genesis of the Abstract Ring Concept." Amer. Math. Monthly 103, 417424, 1996
 Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 2434, 2005
 Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71, 918920, 1964
 Van der Waerden, B. L. A History of Algebra. New York: SpringerVerlag, 1985
 Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002