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Ring (mathematics)
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In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below. However, commonly a ring is defined as a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions; namely the set must be an Abelian group under addition, a monoid under multiplication, such that multiplication distributes over addition.
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In mathematics, a ring is a type of algebraic structure. There is some variation among mathematicians as to exactly what properties a ring is required to have, as described in detail below. However, commonly a ring is defined as a set together with two binary operations (usually called addition and multiplication), where each operation combines two elements to form a third element. To qualify as a ring, the set together with its two operations must satisfy certain conditions; namely the set must be an Abelian group under addition, a monoid under multiplication, such that multiplication distributes over addition. While these operations are familiar from many mathematical structures—such as number systems or the integers, for example—they are also very general in the sense that they take a broad variety of mathematical objects into account. This allows one to handle entities of very different mathematical origins in a flexible way, while retaining essential structural aspects of many objects in abstract algebra and beyond. The ubiquity of rings makes them a central organizing principle of contemporary mathematics. The branch of mathematics that studies rings is known as ring theory.
Definition and illustration
First example: the integers
The most familiar example of a ring is the set of all integers, Z, consisting of the numbers
- ..., -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 under addition; that is:
- Closure axiom for addition: Given two integers a and b, their sum, a + b is also an integer.
- Associativity 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: 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: 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 of addition: For any two integers a and b, a + b = b + a. So the order in which you add two integers is irrelevant.
- The integers form a multiplicative monoid (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 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.
An important difference between the additive and multiplicative properties enumerated above is that multiplicative inverses are not required to exist. For instance, there is no integer a such that a · 2 = 1. This property distinguishes the integers from the rational numbers, which is also a set endowed with operations of addition and multiplication, satisfying all of the above properties but also the existence of a multiplicative inverse for each nonzero element: namely if a and b are nonzero integers the inverse of the rational number a/b is simply b/a. Although in some naive sense the rational numbers are "more complicated" than the integers (e.g. in the sense that a rational number can be viewed as an ordered pair of integers, subject to certain equivalences), in many respects the existence of multiplicative inverses gives the rational numbers a simpler structure. For instance, given a pair of integers a and b we may consider the relation that b is exactly divisible by a, i.e., there exists an integer c such that a · c = b, and using this we get a rich theory of prime and composite numbers, factorization, and so forth. In contrast any nonzero rational number divides any other nonzero rational number, a much less interesting relation. This is prototypical of the different and richer structure that an arbitrary ring has in comparison to a field or a division ring (see below).
Formal definition
A ring is a set R equipped with two binary operations + : R × R ? R and · : R × R ? R (where × denotes the Cartesian product), 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.
| 1. | Closure under addition. | a, b in R, the result of the operation a + b is also in R. | | 2. | Associativity of addition. | For all a, b and 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 under multiplication:
| 1. | Closure under multiplication. | a, b in R, the result of the operation a · b is also in R. | | 2. | Associativity of multiplication. | For all a, b, and 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. |
Commutative ring
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 some a, b in R. Rings that also satisfy commutativity for multiplication are called commutative rings. 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 non-commutative ring is the ring of n × n matrices over a non-trivial field K, for n > 1. In particular, the ring of all 2 × 2 matrices over R (the set of all real numbers) do not form a commutative ring as the following calculation shows:
, which is not equal to
Second example: the ring Z4
Consider the set Z4 consisting of the numbers 0, 1, 2, 3 where addition and multiplication are defined as follows (note that for any integer, x, x mod 4 is defined to be the remainder when x is divided by 4):
- For any x, y in Z4, x + y is defined to be their sum in Z (the set of all integers) mod 4. So we can represent the additive structure of Z4 by the left-most table as shown.
- For any x, y in Z4, 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 Z4 by the right-most 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 Z4 is a ring under these operations. First of all, one can use the left-most table to show that Z4 is closed under addition (any result is either 0, 1, 2 or 3). Associativity of addition in Z4 follows from associativity of addition in the set of all integers. The additive identity is 0 as can be verified by looking at the left-most 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, Z4 is an Abelian group under addition.
Similarly, Z4 is closed under multiplication as the right-most table shows (any result above is either 0, 1, 2 or 3). Associativity of multiplication in Z4 follows from associativity of multiplication in the set of all integers. The multiplicative identity is 1 as can be verified by looking at the right-most table. Therefore, Z4 is a monoid under multiplication.
Distributivity of the two operations over each other follow from distributivity of addition over multiplication (and vice-versa) 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 (Z4, +, ·):
2 · 2 = 0
- although neither factor is 0. In general, a non-zero element a of a ring, (R, +, ·) is said to be a zero divisor in (R, +, ·), if there exists a non-zero 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 Z4 (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
then one can verify that (+, ·) 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 ring (see below).
History
- the founder of ring theory.]]
The study of rings originated from the theory of polynomial rings and the theory of algebraic integers. Furthermore, the appearance of hypercomplex numbers in the mid-nineteenth century undercut the pre-eminence of fields in mathematical analysis.
Richard Dedekind (image to the right) introduced the concept of a ring.
The term ring (Zahlring) was coined by David Hilbert in the article Die Theorie der algebraischen Zahlkörper, Jahresbericht der Deutschen Mathematiker Vereinigung, Vol. 4, 1897. . The motivation for this term is unclear.
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 Noether gave the first axiomatic foundation of the theory of commutative rings in her monumental paper Ideal Theory in Rings.
Simple consequences of the ring axioms
Basic facts about rings that can be deduced from the ring axioms are commonly subsumed under elementary ring theory. For example, one can deduce uniqueness of the additive identity in a ring, and uniqueness of the additive inverse of a particular element in a ring using group theory alone. Similarly, the inverse of a unit in a ring is necessarily unique (provable using group theory). However, some properties that effectively combine the additive and multiplicative structures in a ring cannot be proved by group theoretical means alone. For example, one cannot prove that for every element a in a ring (R, +, ·), a · 0 = 0 · a = 0, using only one of the structures. One also must take advantage of distributivity of multiplication over addition (see below). This constitutes one explanation as to why ring theory is important in its own right. This also explains why distributivity is such an important axiom in ring theory.
Basic properties of rings
From the axioms, one can deduce that if (R, +, ·) is a ring, for all a, b in R we have:
Theorem 1: 0 · a = a · 0 = 0
Corollary 1: A ring, (R, +, ·) is trivial (that is, consists of precisely one element) if and only if 0 = 1.
Theorem 2: (−1)a = −a
Theorem 3: (−a) · b = a · (−b) = −(ab)
Group properties of rings
- An integer in a ring is an element n of the ring that can be written either as:
n = 1 + ... + 1 or,
n = −1 − ... − 1
- Note that for any element a of a ring (R, +, ·), a · n = n · a for any integer n. In words, this means that the order in which you multiply an integer with another element in a ring is irrelevant.
Binomial theorem for rings
- holds whenever x and y commute (see this for a proof). The theorem holds for arbitrary x and y in a commutative ring.
Basic concepts
Zero ring
Let (R, +, ·) be a ring. Then (R, +, ·) is said to be a zero ring if the product of any two elements in R is 0 (the additive identity).
Note
- If we assume that rings have multiplicative identities, then a ring is zero if and only if it is trivial. Therefore, the concept of a zero ring is only interesting when one deals with rings that are only semigroups under multiplication (that is, do not necessarily have a multiplicative identity).
Example
- Any Abelian group can be turned into a zero ring by letting a · b = 0 for every a, b in the group (and letting + be the group operation of that Abelian group).
Units
An element a of a ring (R, +, ·) need not necessarily have a multiplicative inverse (for example, in the ring of all integers, 2 has no multiplicative inverse). An element a in a ring is called a unit if it is invertible with respect to multiplication. Formally,
Formal definition
Let (R, +, ·) be a ring. An element a of (R, +, ·) is said to be a unit in (R, +, ·) if there is an element b in the ring such that a · b = b · a = 1.
Notes
- The set of all units in (R, +, ·) forms a group under ring multiplication; this group is denoted by U(R) or R*.
Examples
- Every non-zero element in the ring of all real numbers ((R, +, ·)) is a unit in this ring. In fact, one can define a field (see below) to be a commutative ring such that every non-zero element is a unit. Therefore, (R, +, ·) forms a field.
Opposite ring
- Let (R, +, ·) be a ring. The opposite ring, (Rop, +, *) is defined by a * b = b · a. Addition on the opposite ring is defined to be the same as addition on the original ring.
Note
- If a ring (R, +, ·) is commutative, then it is isomorphic to its opposite ring. The isomorphism is given by the identity map (in the category of sets).
Examples
- If the opposite ring of a ring (R, +, ·) is commutative, then so is the original ring.
- Consider the opposite ring of the opposite ring; that is Ropop. Note that, Ropop = R so that the operation on rings, taking any ring to its 'opposite' is involutory.
- Note that, in particular, the above example shows that one can define a natural functor from the category of rings to itself.
- If two rings R1 and R2 are isomorphic, then their corresponding opposite rings are also isomorphic.
Ring homomorphisms
A homomorphism of rings is a function f: R1 ? R2 between two rings R1 and R2 preserving the ring operations, in the sense that for all a, b in R1 the following identities are required to hold:
As in any category a map f possessing an inverse g: R2 ? R1, i.e. a map in the opposite direction such that the two compositions f ? g and g ? f equal the identity map of R2 and R1, respectively, is called an isomorphism. Equivalently, f is bijective.
For any ring R with unit element, there is a natural map Z ? R, it maps any positive n to the n-fold sum of the unit element of R, and −n to the additive inverse of . The characteristic of R can be expressed in terms of this map.
Subrings
Informally, a subring is a ring, (S, +, ·), contained in a bigger one, (R, +, ·). More formally, let (R, +, ·) be a ring. A subset S of R is said to be a subring of R if:
- 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, −a (the additive inverse of a in R) is in S
- The multiplicative identity of R is in S, i.e 1 is in S
If S is a subring of R, then S is a ring in its own right with + and · restricted to the cartesian product S X S.
Examples
- Suppose is a ring morphism. Then the image, f (R1) is a subring of R2.
- If A is a subring of R and B is a subring of R such that B is a subset of A, then B is a subring of A.
Ideals
The purpose of an ideal in a ring is to somehow allow one to define the quotient ring of a ring (analogous to the quotient group of a group; see below). An ideal in a ring can therefore be thought of as a generalization of a normal subgroup in a group. More formally, let (R, +, ·) be a ring. A subset 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 ''k'' · ''R'' is a right ideal in ''R'', and ''R'' · ''k'' is a left ideal in ''R''. These ideals (for any ''k'' in ''R'') are called the principal 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.
- An ideal in a ring, (''R'', +, ·), is said to be a ''two-sided ideal'' if it is both a left ideal and right ideal in (''R'', +, ·). It is preferred to call a two-sided ideal, simply an ''ideal''.
- If ''I'' = (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 non-trivial ideals in R (the ring of all real numbers) (i.e, the only ideals in R are and R itself). More generally, a field cannot contain any non-trivial 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 it is a right ideal. So for simplicity's sake, we refer to any ideal in a commutative ring as just an ''ideal''.
Quotient ring
Informally, the quotient ring of a ring, is a generalization of the notion of a quotient group of a group. More formally, given a ring (''R'', +, ·) and an two-sided ideal ''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''.
Direct product of rings
Let ''R'' and ''S'' be rings. Then the product ''R'' X ''S'' can be equipped with the following natural ring structure:
- (''r''1, ''s''1) + (''r''2, ''s''2) = (''r''1 + ''r''2, ''s''1 + ''s''2)
- (''r''1, ''s''1) · (''r''2, ''s''2) = (''r''1 · ''r''2, ''s''1 · ''s''2)
for every ''r''1, ''r''2 in ''R'' and ''s''1, ''s''2 in ''S''. The ring ''R'' X ''S'' with the above operations of addition and operation (derived from ''R'' and ''S'') is called the direct product of ''R'' with ''S''. By repeating this process ''n'' times with ''n'' rings, one can form what is known as the n''-fold product of rings. The product is natural because the quotient ring ''R'' X ''S'' / ''R'' is isomorphic to ''S'' and similarly ''R'' X ''S'' / ''S'' is isomorphic to ''R''.
Center of a ring
There are various notions to address the non-commutativity of general rings ''R''. The ''center'' consists of the ring elements that commute with every other element:
- ''C'' =
It is a subring of ''R''. A subring of ''R'' is said to be ''central'' in ''R'', if it is a subset of ''C'' (and therefore a subring of ''C'').
Boolean rings
A ring ''R'' is called ''Boolean ring'' if every ring element ''a'' is an idempotent, i.e. ''a'' · ''a'' = ''a''. For example, the (unique) ring with two elements has this property. There is a one-to-one correspondence between Boolean algebras and Boolean rings, by expressing set difference and set intersection in terms of addition and multiplication and vice versa. Any Boolean ring is commutative and of characteristic two, i.e. ''a'' + ''a'' = 0 for all ''a'' in ''R''.
Endomorphism rings
Let (''A'', +) be an Abelian group and let ''E'' be the set of all endomorphisms of ''A'' (that is, the set of all group morphisms from ''A'' to itself) . If ''f'' and ''g'' belong to ''E'', then define ''f'' + ''g'' to be the point-wise sum of ''f'' and ''g'' (with respect to the group operation). Similarly, define ''f'' · ''g'' = ''f'' o ''g'' (''f'' composed with ''g''). Then (''E'', +, ·) forms a ring known as the endomorphism ring of (''A'', +).
Example
Ab) to the category of rings using the notion of endomorphism rings.
More generally the endomorphism ring of any module is an important ring when studying the module. Every ring arises as the endomorphism ring of a module, the regular module. Conversely an ''R''-module ''M'' is nothing more than a ring homomorphism from the ring ''R'' into the endomorphism ring of the underlying additive group of ''M''. Properties of modules such as indecomposable and strongly indecomposable are often defined in terms of the module's associated endomorphism ring.
Tensor product of rings
- Since any ring is both a left and right module over itself, it is possible to construct the tensor product of ''R'' over a ring ''S'' with another ring ''T'' to get another ring provided ''S'' is a central subring of ''R'' and ''T''.
Examples and applications
Numbers
Many number systems such as the integers, rational numbers, real numbers and the complex numbers naturally occur as rings. Note that in the case of the rational, real and complex numbers, every non-zero element has a multiplicative inverse. These number systems (except for the integers) form what is known as a field (see below or at the top of the article for a comparison between fields and rings).
Rationals
The desire for the existence of multiplicative inverses in the ring of all integers (see above) suggests considering fractions
-
Fractions of integers (with ''b'' nonzero) are known as rational numbers. The set of all such fractions is commonly denoted by Q. Note that the set of all rationals form a ring under the natural operations of addition and multiplication. In fact, the rationals satisfy a stronger axiom than the integers now that we have included fractions: namely, every ''non-zero rational'' has a multiplicative inverse, its reciprocal, which multiplied by x'' gives the product 1, also known as a reciprocal...
. Because of this, along with the fact that multiplication on the rationals is commutative (see above), the rationals form what is known as a field (see below) . Fields are central objects to abstract algebra.
Nonzero integers modulo a prime
Polynomial rings
- The polynomial ring ''R''[''X''] of polynomials over a ring ''R'' is also a ring.
- The set of formal power series ''R''''X''1, …, ''X''''n'' over a commutative ring ''R'' is a ring.
Some examples of the ubiquity of rings
It is remarkable how many different kinds of mathematical objects can be fruitfully analyzed in terms of some associated ring. For instance:
- To any topological space ''X'' one can associate its integral cohomology ring , a graded ring. As the name suggests, 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 spheres and tori, for which the methods of point-set topology are not well-suited. Cohomology groups were later defined in terms of homology groups in a way which is roughly analogous to the dual of a vector space. To know each individual integral homology group is essentially the same as knowing each individual integral cohomology group, because of the universal coefficient theorem. However, the advantage of the cohomology groups is that there is a natural product, which is analogous to the observation that one can multiply pointwise a ''k''-multilinear form and an ''l''-multilinear form to get a (''k'' + ''l'')-mulilinear form. The importance of the ring structure in cohomology is hard to overstate: it provides the foundation for characteristic classes of fiber bundles, intersection theory on manifolds and algebraic varieties, Schubert calculus and much more.
- To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group 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 ring: 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.
- To any group ring or Hopf algebra is associated its representation ring 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 theory, which is more or less the Grothendieck group given a ring structure.
- To any irreducible algebraic variety is associated its function field. The points of an algebraic variety correspond to valuation rings contained in the function field and containing the coordinate ring. The study of algebraic geometry makes heavy use of commutative algebra to study geometric concepts in terms of ring-theoretic properties. Birational geometry studies maps between the subrings of the function field.
- Every simplicial complex 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 combinatorics. In particular, the algebraic geometry of the Stanley–Reisner ring was used to characterize the numbers of faces in each dimension of simplicial polytopes.
Rings with additional structure
Lie rings
A Lie ring is defined to be a ring that is nonassociative and anticommutative under multiplication, that also satisfies the Jacobi identity. More specifically we can define a Lie ring to be an abelian group 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 groups under addition. Any Lie algebra 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 algebra.
Lie rings are used in the study of finite p-groups through the Lazard correspondence. The lower central factors of a ''p''-group are finite abelian ''p''-groups, 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 commutator of two coset representatives. The Lie ring structure is enriched with another module homomorphism, then ''p''th power map, making the associated Lie ring a so-called restricted Lie ring.
Lie rings are also useful in the definition of a p-adic analytic groups and their endomorphisms by studying Lie algebras over rings of integers such as the p-adic 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 space and (''X'', +, ·) be a ring. Then (''X'', ''T'', +, ·) is said to be a topological ring, 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 topology. So clearly, any topological ring is a topological group (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.
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 non-zero and ''a'' · ''b'' = 0, then ''b'' need not necessarily be 0. In particular, if ''a'' · ''b'' = ''a'' · ''c'' with ''a'' non-zero ''b'' need not equal ''c''. An example of this is the set of ''n'' x ''n'' matrices over R where ''a'' maybe such a non-zero 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 non-trivial commutative ring with no zero divisors). But we still run into a problem; namely we can't necessarily divide by non-zero 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 field is a ring in which the non-zero elements form an abelian group 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 field, if its non-zero 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 non-zero 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:
Fields and integral domains are very important in modern algebra.
Category theoretical description
Note that every ring is an Abelian group and every morphism of rings is a morphism of the underlying Abelian groups. Therefore, rings can be thought of as monoids in Ab, the category of abelian groups (thought of as a monoidal category under the tensor product). The monoid action of a ring ''R'' on a abelian group is simply an ''R''-module.
It follows that a ring may be regarded as a preadditive category (a category enriched over Ab) with a single object. Here the morphisms are the ring elements, composition of morphisms is ring multiplication, and the additive structure on morphisms is ring addition. The opposite ring is then the categorical dual.
See also
Citations
General references
- Atiyah M. F., Macdonald, I. G., ''Introduction to commutative algebra''. Addison-Wesley Publishing Co., Reading, Mass.-London-Don 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."
- 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 0-521-36086-2
- 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 0-88385-015-X
- Nagell, T. "Moduls, Rings, and Fields." §6 in Introduction to Number Theory. New York: Wiley, pp. 19-21, 1951
- Nathan Jacobson, ''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.
| year=1974}}
- Lam, T. Y., ''A first course in noncommutative rings''. Second edition. Graduate Texts in Mathematics, 131. Springer-Verlag, New York, 2001. xx+385 pp. ISBN 0-387-95183-0
- Lam, T. Y., ''Exercises in classical ring theory''. Second edition. Problem Books in Mathematics. Springer-Verlag, New York, 2003. xx+359 pp. ISBN 0-387-00500-5
- Lam, T. Y., ''Lectures on modules and rings''. Graduate Texts in Mathematics, 189. Springer-Verlag, New York, 1999. xxiv+557 pp. ISBN 0-387-98428-3
.
- 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 0-8218-2169-5
- Rowen, Louis H., ''Ring theory''. Vol. I, II. Pure and Applied Mathematics, 127, 128. Academic Press, Inc., Boston, MA, 1988. ISBN 0-12-599841-4, ISBN 0-12-599842-2
- Sloane, N. J. A. Sequences A027623 and A037234 in "The On-Line 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, 222-227, 1947.
- Berrick, A. J. and Keating, M. E. An Introduction to Rings and Modules with K-Theory in View. Cambridge, England: Cambridge University Press, 2000.
| year=1995 | volume=150}}
- Fine, B. "Classification of Finite Rings of Order ." Math. Mag. 66, 248-252, 1993
- Fletcher, C. R. "Rings of Small Order." Math. Gaz. 64, 9-22, 1980
- Fraenkel, A. "Über die Teiler der Null und die Zerlegung von Ringen." J. reine angew. Math. 145, 139-176, 1914
- Gilmer, R. and Mott, J. "Associative Rings of Order ." Proc. Japan Acad. 49, 795-799, 1973
- Harris, J. W. and Stocker, H. Handbook of Mathematics and Computational Science. New York: Springer-Verlag, 1998
- Knuth, D. E. The Art of Computer Programming, Vol. 2: Seminumerical Algorithms, 3rd ed. Reading, MA: Addison-Wesley, 1998
- Korn, G. A. and Korn, T. M. Mathematical Handbook for Scientists and Engineers. New York: Dover, 2000
| year=1962 | volume=13 | pages=xiii+234}}
- Pierce, Richard S., ''Associative algebras''. Graduate Texts in Mathematics, 88. Studies in the History of Modern Science, 9. Springer-Verlag, New York-Berlin, 1982. xii+436 pp. ISBN 0-387-90693-2
Historical references
- 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: Springer-Verlag, 2004. ISBN 3-540-43491-7
- 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 0-8218-0993-8
- 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, 417-424, 1996
- Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005
- Singmaster, D. and Bloom, D. M. "Problem E1648." Amer. Math. Monthly 71, 918-920, 1964
- Van der Waerden, B. L. A History of Algebra. New York: Springer-Verlag, 1985
- Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002
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