List of abstract algebra topics - AbsoluteAstronomy.com

Abstract algebra
Abstract algebra
Abstract algebra is the subject area of mathematics that studies algebraic structures, such as groups, rings, fields, modules, vector spaces, and algebras...

is the subject area of mathematics

Mathematics

Mathematics 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...

that studies algebraic structure

Algebraic structure

In 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...

s, such as groups

Group (mathematics)

In 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...

, rings

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

, fields

Field (mathematics)

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...

, modules

Module (mathematics)

In 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...

, vector space

Vector space

A 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, and algebras

Algebra over a field

In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...

. The phrase abstract algebra was coined at the turn of the 20th century to distinguish this area from what was normally referred to as algebra, the study of the rules for manipulating formulae and algebraic expressions involving unknowns and real or complex number

Complex number

A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...

s, often now called elementary algebra
Elementary algebra
Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. It is typically taught in secondary school under the term algebra. The major difference between algebra and...

. The distinction is rarely made in more recent writings.

Basic language

Unary operator
Binary operation
Binary operation
In 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....
- Multiplication table
  Multiplication table
  In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system....
Closure (mathematics)
Closure (mathematics)
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...
Arity
Arity
In logic, mathematics, and computer science, the arity of a function or operation is the number of arguments or operands that the function takes. The arity of a relation is the dimension of the domain in the corresponding Cartesian product...
Commutative operation
Identity element
Identity element
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...
Additive inverse
Additive inverse
In 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....

, multiplicative inverse
Multiplicative inverse
In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For the multiplicative inverse of a real number, divide 1 by the...

, inverse element
Inverse element
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...
Cancellation property
Cancellation property
In mathematics, the notion of cancellative is a generalization of the notion of invertible.An element a in a magma has the left cancellation property if for all b and c in M, a * b = a * c always implies b = c.An element a in a magma has the right cancellation...
Associativity
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...
Distributivity
Distributivity
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:...

Semigroups and monoids

Semigroup
Semigroup
In 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...
- Free semigroup
  Free semigroup
  In abstract algebra, the free monoid on a set A is the monoid whose elements are all the finite sequences of zero or more elements from A. It is usually denoted A∗. The identity element is the unique sequence of zero elements, often called the empty string and denoted by ε or λ, and the...
- Green's relations
  Green's relations
  In mathematics, Green's relations are five equivalence relations that characterise the elements of a semigroup in terms of the principal ideals they generate. The relations are named for James Alexander Green, who introduced them in a paper of 1951...
- Krohn–Rhodes theory
- Transformation semigroup
  Transformation semigroup
  In algebra and theoretical computer science, an action or act of a semigroup on a set is a rule which associates to each element of the semigroup a transformation of the set in such a way that the product of two elements of the semigroup is associated with the composite of the two corresponding...
- Semigroupoid
  Semigroupoid
  In mathematics, a semigroupoid is a partial algebra which satisfies the axioms for a small category, except possibly for the requirement that there be an identity at each object...
- Subsemigroup
Monoid
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...
- Aperiodic monoid
  Aperiodic monoid
  In mathematics, an aperiodic semigroup is a semigroup S such that for every x ∈ S, there exists a nonnegative integer n such thatxn = xn + 1.An aperiodic monoid is an aperiodic semigroup which is a monoid...
Coimage
Coimage
In algebra, the coimage of a homomorphismis the quotientof domain and kernel.The coimage is canonically isomorphic to the image by the first isomorphism theorem, when that theorem applies....

Group theory

Group (mathematics)
Group (mathematics)
In 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...
Finite group
Finite group
In 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...
Coset
Coset
In mathematics, if G is a group, and H is a subgroup of G, and g is an element of G, thenA coset is a left or right coset of some subgroup in G...
Lagrange's theorem (group theory)
Lagrange's theorem (group theory)
Lagrange's theorem, in the mathematics of group theory, states that for any finite group G, the order of every subgroup H of G divides the order of G. The theorem is named after Joseph Lagrange....
Abelian group
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...
- Torsion subgroup
  Torsion subgroup
  In the theory of abelian groups, the torsion subgroup AT of an abelian group A is the subgroup of A consisting of all elements that have finite order...
- Direct sum of groups
- Free abelian group
  Free abelian group
  In 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...
- Finitely generated abelian group
  Finitely generated abelian group
  In abstract algebra, an abelian group is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the formwith integers n1,...,ns...
- Rank of an abelian group
  Rank of an abelian group
  In mathematics, the rank, Prüfer rank, or torsion-free rank of an abelian group A is the cardinality of a maximal linearly independent subset. The rank of A determines the size of the largest free abelian group contained in A. If A is torsion-free then it embeds into a vector space over the...
- Divisible group
  Divisible group
  In mathematics, especially in the field of group theory, a divisible group is an abelian group in which every element can, in some sense, be divided by positive integers, or more accurately, every element is an nth multiple for each positive integer n...
- Locally cyclic group
  Locally cyclic group
  In group theory, a locally cyclic group is a group in which every finitely generated subgroup is cyclic.-Some facts:*Every cyclic group is locally cyclic, and every locally cyclic group is abelian....
Cyclic group
Cyclic group
In 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...
Klein four-group
Klein four-group
In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
Elementary group theory
- Quaternion group
  Quaternion group
  In group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
- List of small groups
- Examples of groups
  Examples of groups
  Some elementary examples of groups in mathematics are given on Group .Further examples are listed here.-Permutations of a set of three elements:Consider three colored blocks , initially placed in the order RGB...
Normal subgroup
Normal subgroup
In 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....
- Conjugacy class
  Conjugacy class
  In mathematics, especially group theory, the elements of any group may be partitioned into conjugacy classes; members of the same conjugacy class share many properties, and study of conjugacy classes of non-abelian groups reveals many important features of their structure...
  - Inner automorphism
    Inner automorphism
    In abstract algebra an inner automorphism is a functionwhich, informally, involves a certain operation being applied, then another one performed, and then the initial operation being reversed...
  - Conjugate closure
    Conjugate closure
    In group theory, the conjugate closure of a subset S of a group G is the subgroup of G generated by SG, i.e. the closure of SG under the group operation, where SG is the conjugates of the elements of S:The conjugate closure of S is denoted or G.The conjugate closure of any subset S of a group G...
- Semidirect product
  Semidirect product
  In mathematics, specifically in the area of abstract algebra known as group theory, a semidirect product is a particular way in which a group can be put together from two subgroups, one of which is a normal subgroup. A semidirect product is a generalization of a direct product...
- Dihedral group
  Dihedral group
  In mathematics, a dihedral group is the group of symmetries of a regular polygon, including both rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, geometry, and chemistry.See also: Dihedral symmetry in three...
- Dicyclic group
- Extension problem
- Hamiltonian group
  Hamiltonian group
  In group theory, a Dedekind group is a group G such that every subgroup of G is normal.All abelian groups are Dedekind groups.A non-abelian Dedekind group is called a Hamiltonian group....
- Simple group
  Simple group
  In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself. A group that is not simple can be broken into two smaller groups, a normal subgroup and the quotient group, and the process can be repeated...
  - Classification of finite simple groups
    Classification of finite simple groups
    In 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...
- Composition series
  Composition series
  In abstract algebra, a composition series provides a way to break up an algebraic structure, such as a group or a module, into simple pieces. The need for considering composition series in the context of modules arises from the fact that many naturally occurring modules are not semisimple, hence...
- Solvable group
  Solvable group
  In mathematics, more specifically in the field of group theory, a solvable group is a group that can be constructed from abelian groups using extensions...
- Nilpotent group
  Nilpotent group
  In mathematics, more specifically in the field of group theory, a nilpotent group is a group that is "almost abelian". This idea is motivated by the fact that nilpotent groups are solvable, and for finite nilpotent groups, two elements having relatively prime orders must commute...
Characteristic subgroup
Characteristic subgroup
In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
Derived group
Centralizer and normalizer
Centralizer and normalizer
In group theory, the centralizer and normalizer of a subset S of a group G are subgroups of G which have a restricted action on the elements of S and S as a whole, respectively...
Automorphism group
Sylow theorems
- Local analysis
  Local analysis
  In mathematics, the term local analysis has at least two meanings - both derived from the idea of looking at a problem relative to each prime number p first, and then later trying to integrate the information gained at each prime into a 'global' picture...
Group action
Group action
In 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...
- Permutation group
  Permutation group
  In mathematics, a permutation group is a group G whose elements are permutations of a given set M, and whose group operation is the composition of permutations in G ; the relationship is often written as...
- Symmetric group
  Symmetric group
  In mathematics, the symmetric group Sn on a finite set of n symbols is the group whose elements are all the permutations of the n symbols, and whose group operation is the composition of such permutations, which are treated as bijective functions from the set of symbols to itself...
- Alternating group
- Point group
  Point group
  In geometry, a point group is a group of geometric symmetries that keep at least one point fixed. Point groups can exist in a Euclidean space with any dimension, and every point group in dimension d is a subgroup of the orthogonal group O...
- Stabilizer subgroup
- Orbit (group theory)
- Orbit-stabilizer theorem
- Cayley's theorem
  Cayley's theorem
  In group theory, Cayley's theorem, named in honor of Arthur Cayley, states that every group G is isomorphic to a subgroup of the symmetric group acting on G...
- Burnside's lemma
  Burnside's lemma
  Burnside's lemma, sometimes also called Burnside's counting theorem, the Cauchy-Frobenius lemma or the orbit-counting theorem, is a result in group theory which is often useful in taking account of symmetry when counting mathematical objects. Its various eponyms include William Burnside, George...
- Wreath product
  Wreath product
  In mathematics, the wreath product of group theory is a specialized product of two groups, based on a semidirect product. Wreath products are an important tool in the classification of permutation groups and also provide a way of constructing interesting examples of groups.Given two groups A and H...
Free group
Free group
In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many elements of S and their inverses...
Presentation of a group
Presentation of a group
In mathematics, one method of defining a group is by a presentation. One specifies a set S of generators so that every element of the group can be written as a product of powers of some of these generators, and a set R of relations among those generators...
Free product
Free product
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new group G ∗ H. The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “most general” group having these properties...
Word problem for groups
Word problem for groups
In mathematics, especially in the area of abstract algebra known as combinatorial group theory, the word problem for a finitely generated group G is the algorithmic problem of deciding whether two words in the generators represent the same element...
Burnside's problem
Burnside's problem
The Burnside problem, posed by William Burnside in 1902 and one of the oldest and most influential questions in group theory, asks whether a finitely generated group in which every element has finite order must necessarily be a finite group...

Ring theory

Ring (mathematics)
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
Subring
Subring
In 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...
Ring homomorphism
Ring homomorphism
In ring theory or abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication....
- Ring epimorphism
- Ring monomorphism
- Ring isomorphism
Ring ideal
- Principal ideal
  Principal ideal
  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:...
- Ideal quotient
- Maximal ideal
  Ideal (ring theory)
  In 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"....
  
  , prime ideal
  Prime ideal
  In algebra , a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers...
- Jacobson radical
  Jacobson radical
  In 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 R-modules. It happens that substituting "left" in place of "right" in the definition yields the same...
- Radical of an ideal
  Radical of an ideal
  In commutative ring theory, a branch of mathematics, the radical of an ideal I is an ideal such that an element x is in the radical if some power of x is in I. A radical ideal is an ideal that is its own radical...
Quotient ring
Quotient ring
In 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...
Simple ring
Simple ring
In abstract algebra, a simple ring is a non-zero 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...
Product of rings
Product of rings
In mathematics, it is possible to combine several rings into one large product ring. This is done as follows: if I is some index set and Ri is a ring for every i in I, then the cartesian product Πi in I Ri can be turned into a ring by defining the operations coordinatewise, i.e...
Unit (ring theory)
Unit (ring theory)
In 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...
Order (ring theory)
Idempotent
Nilpotent
Nilpotent
In mathematics, an element x of a ring R is called nilpotent if there exists some positive integer n such that xn = 0....

, reduced ring
Reduced ring
In ring theory, a ring R is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, x2 = 0 implies x = 0...
Zero divisor
Zero divisor
In 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...
Integral domain
Euclidean domain
Euclidean domain
In 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...
Principal ideal domain
Principal ideal domain
In 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...
Domain (ring theory)
Domain (ring theory)
In mathematics, especially in the area of abstract algebra known as ring theory, a domain is a ring such that ab = 0 implies that either a = 0 or b = 0. That is, it is a ring which has no left or right zero divisors. Some authors require the ring to be nontrivial...
Prime ring
Prime ring
In abstract algebra, a non-trivial ring R is a prime ring if for any two elements a and b of R, arb = 0 for all r in R implies that either a = 0 or b = 0. Prime ring can also refer to the subring of a field determined by its characteristic...
Primitive ring
Primitive ring
In the branch of abstract algebra known as ring theory, a left primitive ring is a ring which has a faithful simple left module. Well known examples include endomorphism rings of vector spaces and Weyl algebras over fields of characteristic zero.- Definition :...
Primitive ideal
Primitive ideal
In mathematics, a left primitive ideal in ring theory is the annihilator of a simple left module. A right primitive ideal is defined similarly. Note that left and right primitive ideals are always two-sided ideals....
Semiprimitive ring
Semiprimitive ring
In mathematics, especially in the area of algebra known as ring theory, a semiprimitive ring is a type of ring more general than a semisimple ring, but where simple modules still provide enough information about the ring. Important rings such as the ring of integers are semiprimitive, and an...
Matrix ring
Matrix ring
In abstract algebra, a matrix ring is any collection of matrices forming a ring under matrix addition and matrix multiplication. The set of n×n matrices with entries from another ring is a matrix ring, as well as some subsets of infinite matrices which form infinite matrix rings...
Jacobson density theorem
Jacobson density theorem
In mathematics, more specifically non-commutative ring theory, modern algebra, and module theory, the Jacobson density theorem is a theorem concerning simple modules over a ring R....
Field (mathematics)
Field (mathematics)
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...
Field of fractions
Field of fractions
In abstract algebra, the field of fractions or field of quotients of an integral domain is the smallest field in which it can be embedded. The elements of the field of fractions of the integral domain R have the form a/b with a and b in R and b ≠ 0...
Commutative algebra
Commutative algebra
Commutative 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...
Polynomial
Polynomial
In mathematics, a polynomial is an expression of finite length constructed from variables and constants, using only the operations of addition, subtraction, multiplication, and non-negative integer exponents...

, monomial
Monomial
In mathematics, in the context of polynomials, the word monomial can have one of two different meanings:*The first is a product of powers of variables, or formally any value obtained by finitely many multiplications of a variable. If only a single variable x is considered, this means that any...
Polynomial ring
Polynomial ring
In 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...

, symmetric algebra
Symmetric algebra
In mathematics, the symmetric algebra S on a vector space V over a field K is the free commutative unital associative algebra over K containing V....
Center (algebra)
Center (algebra)
The term center or centre is used in various contexts in abstract algebra to denote the set of all those elements that commute with all other elements. It is often denoted Z, from German Zentrum, meaning "center". More specifically:...
Integral closure
Dedekind domain
Dedekind domain
In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessarily unique up to the order of the factors...
- Ideal class group
  Ideal class group
  In mathematics, the extent to which unique factorization fails in the ring of integers of an algebraic number field can be described by a certain group known as an ideal class group...
Unique factorization domain
Unique factorization domain
In 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...
Formal power series
Formal power series
In mathematics, formal power series are a generalization of polynomials as formal objects, where the number of terms is allowed to be infinite; this implies giving up the possibility to substitute arbitrary values for indeterminates...
Characteristic (algebra)
Characteristic (algebra)
In mathematics, the characteristic of a ring R, often denoted char, is defined to be the smallest number of times one must use the ring's multiplicative identity element in a sum to get the additive identity element ; the ring is said to have characteristic zero if this repeated sum never reaches...
Boolean algebra (structure), Boolean ring
Boolean ring
In 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....

, Boolean homomorphism
Division ring
Division ring
In abstract algebra, a division ring, also called a skew field, is a ring in which division is possible. Specifically, it is a non-trivial ring in which every non-zero element a has a multiplicative inverse, i.e., an element x with...

, division algebra
Division algebra
In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field, in which division is possible.- Definitions :...
Quaternions
- Hurwitz quaternion
  Hurwitz quaternion
  In mathematics, a Hurwitz quaternion is a quaternion whose components are either all integers or all half-integers...
Local ring
Local ring
In abstract algebra, more particularly in ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic number fields examined at a particular place, or...
- Discrete valuation ring
  Discrete valuation ring
  In abstract algebra, a discrete valuation ring is a principal ideal domain with exactly one non-zero maximal ideal.This means a DVR is an integral domain R which satisfies any one of the following equivalent conditions:...
- Regular local ring
  Regular local ring
  In commutative algebra, a regular local ring is a Noetherian local ring having the property that the minimal number of generators of its maximal ideal is equal to its Krull dimension. In symbols, let A be a Noetherian local ring with maximal ideal m, and suppose a1, ..., an is a minimal set of...
Valuation (mathematics)
Semi-local ring
Semi-local ring
In mathematics, a semi-local ring is a ring for which R/J is a semisimple ring, where J is the Jacobson radical of R. The above definition implies R has a finite number of maximal right ideals...
Algebra (ring theory)
Algebra (ring theory)
In mathematics, specifically in ring theory, an algebra over a commutative ring is a generalization of the concept of an algebra over a field, where the base field K is replaced by a commutative ring R....
Cohen–Macaulay ring
Gorenstein ring
Gorenstein ring
In commutative algebra, a Gorenstein local ring is a Noetherian commutative local ring R with finite injective dimension, as an R-module. There are many equivalent conditions, some of them listed below, most dealing with some sort of duality condition....
Monoid ring
Group ring
Group ring
In 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 one-to-one with the given group. As a ring, its addition law is that of the free...
Group algebra
Group algebra
In mathematics, the group algebra is any of various constructions to assign to a locally compact group an operator algebra , such that representations of the algebra are related to representations of the group...
Dual number
Dual number
In linear algebra, the dual numbers extend the real numbers by adjoining one new element ε with the property ε2 = 0 . The collection of dual numbers forms a particular two-dimensional commutative unital associative algebra over the real numbers. Every dual number has the form z = a + bε with a and...
Differential algebra
Differential algebra
In mathematics, differential rings, differential fields, and differential algebras are rings, fields, and algebras equipped with a derivation, which is a unary function that is linear and satisfies the Leibniz product law...
Artinian ring
Artinian ring
In abstract algebra, an Artinian ring is a ring that satisfies the descending chain condition on ideals. They are also called Artin rings and are named after Emil Artin, who first discovered that the descending chain condition for ideals simultaneously generalizes finite rings and rings that are...
Noetherian ring
Noetherian ring
In 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 non-empty set of ideals has a maximal element...
- Hilbert's basis theorem
  Hilbert's basis theorem
  In mathematics, specifically commutative algebra, Hilbert's basis theorem states that every ideal in the ring of multivariate polynomials over a Noetherian ring is finitely generated. This can be translated into algebraic geometry as follows: every algebraic set over a field can be described as the...
Von Neumann regular ring
Von Neumann regular ring
In mathematics, a von Neumann regular ring is a ring R such that for every a in R there exists an x in R withOne may think of x as a "weak inverse" of a...
Banach algebra
Banach algebra
In mathematics, especially functional analysis, a Banach algebra, named after Stefan Banach, is an associative algebra A over the real or complex numbers which at the same time is also a Banach space...
Associative algebra
Associative algebra
In 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...
- Subalgebra
  Subalgebra
  In mathematics, the word "algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation. Algebras in universal algebra are far more general: they are a common generalisation of all algebraic structures...
- Algebra over a field
  Algebra over a field
  In mathematics, an algebra over a field is a vector space equipped with a bilinear vector product. That is to say, it isan algebraic structure consisting of a vector space together with an operation, usually called multiplication, that combines any two vectors to form a third vector; to qualify as...
- Simple algebra
- Central simple algebra
  Central simple algebra
  In ring theory and related areas of mathematics a central simple algebra over a field K is a finite-dimensional associative algebra A, which is simple, and for which the center is exactly K...
- Artin–Wedderburn theorem
  Artin–Wedderburn theorem
  In abstract algebra, the Artin–Wedderburn theorem is a classification theorem for semisimple rings. The theorem states that an Artinian semisimple ring R is isomorphic to a product of finitely many ni-by-ni matrix rings over division rings Di, for some integers ni, both of which are uniquely...
- Skolem–Noether theorem
  Skolem–Noether theorem
  In mathematics, the Skolem–Noether theorem, named after Thoralf Skolem and Emmy Noether, is an important result in ring theory which characterizes the automorphisms of simple rings....
- Tensor algebra
  Tensor algebra
  In mathematics, the tensor algebra of a vector space V, denoted T or T•, is the algebra of tensors on V with multiplication being the tensor product...
  
  , free algebra
  Free algebra
  In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring .-Definition:...
- Tensor product of R-algebras
- Brauer group
  Brauer group
  In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras. It arose out of attempts to classify division algebras over a field and is...
Hypercomplex number
Hypercomplex number
In 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...
- Graded algebra
  Graded algebra
  In mathematics, in particular abstract algebra, a graded algebra is an algebra over a field with an extra piece of structure, known as a gradation ....
Rig (algebra)
Nearring
Nearring
In mathematics, a near-ring is an algebraic structure similar to a ring, but that satisfies fewer axioms. Near-rings arise naturally from functions on groups.- Definition :...

Field theory

Subfield (mathematics)
Field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
Multiplicative group
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
Algebraic number field
Algebraic number field
In mathematics, an algebraic number field F is a finite field extension of the field of rational numbers Q...
Global field
Global field
In mathematics, the term global field refers to either of the following:*an algebraic number field, i.e., a finite extension of Q, or*a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of Fq, the field of rational functions...
Local field
Local field
In mathematics, a local field is a special type of field that is a locally compact topological field with respect to a non-discrete topology.Given such a field, an absolute value can be defined on it. There are two basic types of local field: those in which the absolute value is archimedean and...
Finite field
Finite field
In abstract algebra, a finite field or Galois field is a field that contains a finite number of elements. Finite fields are important in number theory, algebraic geometry, Galois theory, cryptography, and coding theory...
Algebraic element
Algebraic element
In mathematics, if L is a field extension of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g with coefficients in K such that g=0...
Algebraic extension
Algebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
Symmetric function
Symmetric function
In algebra and in particular in algebraic combinatorics, the ring of symmetric functions, is a specific limit of the rings of symmetric polynomials in n indeterminates, as n goes to infinity...
Splitting field
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
Separable polynomial
Separable polynomial
In mathematics, two slightly different notions of separable polynomial are used, by different authors.According to the most common one, a polynomial P over a given field K is separable if all its roots are distinct in an algebraic closure of K, that is the number of its distinct roots is equal to...
Separable extension
Separable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
Primitive element (field theory)
Algebraic closure
Algebraic closure
In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics....
Algebraically closed field
Algebraically closed field
In mathematics, a field F is said to be algebraically closed if every polynomial with one variable of degree at least 1, with coefficients in F, has a root in F.-Examples:...
Transcendence degree
Transcendence degree
In abstract algebra, the transcendence degree of a field extension L /K is a certain rather coarse measure of the "size" of the extension...
Normal extension
Normal extension
In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]...
Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
- Galois group
  Galois group
  In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
- Conjugate element (field theory)
  Conjugate element (field theory)
  In mathematics, in particular field theory, the conjugate elements of an algebraic element α, over a field K, are the roots of the minimal polynomialof α over K.-Example:The cube roots of the number one are:...
- Abelian extension
  Abelian extension
  In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is a cyclic group, we have a cyclic extension. More generally, a Galois extension is called solvable if its Galois group is solvable....
- Kummer theory
  Kummer theory
  In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...
Field norm
Field norm
In mathematics, the norm is a mapping defined in field theory, to map elements of a larger field into a smaller one.-Formal definitions:1. Let K be a field and L a finite extension of K...
Field trace
Field trace
In mathematics, the field trace is a function defined with respect to a finite field extension L/K. It is a K-linear map from L to K...
Formally real field
Formally real field
In mathematics, in particular in field theory and real algebra, a formally real field is a field that admits an ordering which makes it an ordered field.-Alternative Definitions:...
Real closed field
Real closed field
In mathematics, a real closed field is a field F that has the same first-order properties as the field of real numbers. Some examples are the field of real numbers, the field of real algebraic numbers, and the field of hyperreal numbers.-Definitions:...
Tensor product of fields
Tensor product of fields
In abstract algebra, the theory of fields lacks a direct product: the direct product of two fields, considered as a ring is never itself a field. On the other hand it is often required to 'join' two fields K and L, either in cases where K and L are given as subfields of a larger field M, or when K...
Inverse Galois problem
Inverse Galois problem
In Galois theory, the inverse Galois problem concerns whether or not every finite group appears as the Galois group of some Galois extension of the rational numbers Q. This problem, first posed in the 19th century, is unsolved....

Module theory

Free module
Free module
In mathematics, a free module is a free object in a category of modules. Given a set S, a free module on S is a free module with basis S.Every vector space is free, and the free vector space on a set is a special case of a free module on a set.-Definition:...
Finitely-generated module
Finitely-generated module
In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated R-module also may be called a finite R-module or finite over R....
Finitely-presented module
Projective module
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
- Swan's theorem
  Swan's theorem
  In the mathematical fields of topology and K-theory, the Serre–Swan theorem, also called Swan's theorem, relates the geometric notion of vector bundles to the algebraic concept of projective modules and gives rise to a common intuition throughout mathematics: "projective modules over commutative...
- Quillen–Suslin theorem
  Quillen–Suslin theorem
  The Quillen–Suslin theorem, also known as Serre's problem or Serre's conjecture, is a theorem in commutative algebra about the relationship between free modules and projective modules over polynomial rings...
Injective module
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
- Injective hull
  Injective hull
  In mathematics, especially in the area of abstract algebra known as module theory, the injective hull of a module is both the smallest injective module containing it and the largest essential extension of it...
Flat module
Flat module
In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
Direct sum
Direct sum of modules
In abstract algebra, the direct sum is a construction which combines several modules into a new, larger module. The result of the direct summation of modules is the "smallest general" module which contains the given modules as submodules...
Simple module
Simple module
In mathematics, specifically in ring theory, the simple modules over a ring R are the modules over R which have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a non-zero element of M equals M...
Indecomposable module
Indecomposable module
In abstract algebra, a module is indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules.Indecomposable is a weaker notion than simple module:simple means "no proper submodule" N...
Pure submodule
Pure submodule
In mathematics, especially in the field of module theory, the concept of pure submodule provides a generalization of direct summand, a type of particularly well-behaved piece of a module. Pure modules are complementary to flat modules and generalize Prüfer's notion of pure subgroups...
Algebraically compact module
Algebraically compact module
In mathematics, especially in the area of abstract algebra known as module theory, algebraically compact modules, also called pure-injective modules, are modules that have a certain "nice" property which allows the solution of infinite systems of equations in the module by finitary means. The...
Bimodule
Bimodule
In abstract algebra a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible...
Annihilator (ring theory)
Annihilator (ring theory)
In 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 R-module. Choose a nonempty subset S of M...
Regular sequence (algebra)
Regular sequence (algebra)
In commutative algebra, if R is a commutative ring and M an R-module, a nonzero element r in R is called M-regular if r is not a zerodivisor on M, and M/rM is nonzero...

, depth (algebra)
Localization of a module
Fractional ideal
Fractional ideal
In mathematics, in particular commutative algebra, the concept of fractional ideal is introduced in the context of integral domains and is particularly fruitful in the study of Dedekind domains. In some sense, fractional ideals of an integral domain are like ideals where denominators are allowed...
Length of a module
Length of a module
In abstract algebra, the length of a module is a measure of the module's "size". It is defined to be the length of the longest chain of submodules and is a generalization of the concept of dimension for vector spaces...
Fitting lemma
Fitting lemma
The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. Suppose M is a module over some ring...
Artinian module
Artinian module
In abstract algebra, an Artinian module is a module that satisfies the descending chain condition on its poset of submodules. They are for modules what Artinian rings are for rings, and a ring is Artinian if and only if it is an Artinian module over itself...

Representation theory

Algebra representation
Group representation
Group representation
In the mathematical field of representation theory, group representations describe abstract groups in terms of linear transformations of vector spaces; in particular, they can be used to represent group elements as matrices so that the group operation can be represented by matrix multiplication...
Lie algebra representation
Maschke's theorem
Maschke's theorem
In mathematics, Maschke's theorem, named after Heinrich Maschke, is a theorem in group representation theory that concerns the decomposition of representations of a finite group into irreducible pieces...
Schur's lemma
Schur's lemma
In mathematics, Schur's lemma is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if M and N are two finite-dimensional irreducible representations...
Equivariant map
Affine representation
Affine representation
An affine representation of a topological group G on an affine space A is a continuous group homomorphism from G to the automorphism group of A, the affine group Aff...
Projective representation
Projective representation
In the mathematical field of representation theory, a projective representation of a group G on a vector space V over a field F is a group homomorphism from G to the projective linear groupwhere GL is the general linear group of invertible linear transformations of V over F and F* here is the...
Modular representation theory
Modular representation theory
Modular representation theory is a branch of mathematics, and that part of representation theory that studies linear representations of finite group G over a field K of positive characteristic...
Quiver (mathematics)
Quiver (mathematics)
In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation, V, of a quiver assigns a vector space V to each vertex x of the quiver and a linear map V to each...
Representation theory of Hopf algebras

Non-associative systems

Nonassociative ring
Nonassociative ring
In 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...
Associator
Associator
In abstract algebra, the term associator is used in different ways as a measure of the nonassociativity of an algebraic structure.-Ring theory:...
Lie algebra
Lie algebra
In 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...

(see also list of Lie group topics and list of representation theory topics)
- Jacobi identity
  Jacobi identity
  In 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...
- Superalgebra
  Superalgebra
  In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading....
- Universal enveloping algebra
  Universal enveloping algebra
  In mathematics, for any Lie algebra L one can construct its universal enveloping algebra U. This construction passes from the non-associative 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...
- Virasoro algebra
  Virasoro algebra
  In mathematics, the Virasoro algebra is a complex Lie algebra, given as a central extension of the complex polynomial vector fields on the circle, and is widely used in conformal field theory and string theory....
Jordan algebra
Jordan algebra
In abstract algebra, a Jordan algebra is an algebra over a field whose multiplication satisfies the following axioms:# xy = yx # = x ....
Magma (algebra)
Magma (algebra)
In abstract algebra, a magma is a basic kind of algebraic structure. Specifically, a magma consists of a set M equipped with a single binary operation M \times M \rightarrow M....
Loop (algebra)
Quasigroup
Quasigroup
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
Alternative algebra
Alternative algebra
In abstract algebra, an alternative algebra is an algebra in which multiplication need not be associative, only alternative. That is, one must have*x = y*x = y...
Power associativity
Power associativity
In abstract algebra, power associativity is a property of a binary operation which is a weak form of associativity.An algebra is said to be power-associative if the subalgebra generated by any element is associative....
Cayley–Dickson construction
- Octonions
- Sedenions

Generalities

Algebraic structure
Algebraic structure
In 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...
Universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
- Mathematical variety
  Mathematical variety
  In mathematics the meaning of variety can be* in algebraic geometry, an algebraic variety, which may be affine, projective or abstractor...
- Congruence relation
  Congruence relation
  In abstract algebra, a congruence relation is an equivalence relation on an algebraic structure that is compatible with the structure...
- Free object
  Free object
  In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. It is a part of universal algebra, in the sense that it relates to all types of algebraic structure . It also has a formulation in terms of category theory, although this is in yet more abstract terms....
- Generating set
  Generating set
  In mathematics, the expressions generator, generate, generated by and generating set can have several closely related technical meanings:...
- Clone (algebra)
  Clone (algebra)
  In universal algebra, a clone is a set C of operations on a set A such that*C contains all the projections , defined by ,*C is closed under composition : if f, g1, …, gm are members of C such that f is m-ary, and gj is n-ary for every j, then the n-ary operation is in C.Given an algebra...
Kernel of a function
Kernel of a function
In set theory, the kernel of a function f may be taken to be either*the equivalence relation on the function's domain that roughly expresses the idea of "equivalent as far as the function f can tell", or*the corresponding partition of the domain....
- Kernel (algebra)
  Kernel (algebra)
  In the various branches of mathematics that fall under the heading of abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective. An important special case is the kernel of a matrix, also called the null space.The definition of kernel takes...
- Isomorphism class
  Isomorphism class
  An isomorphism class is a collection of mathematical objects isomorphic to each other.Isomorphism classes are often defined if the exact identity of the elements of the set is considered irrelevant, and the properties of the structure of the mathematical object are studied. Examples of this are...
- Isomorphism theorem
  Isomorphism theorem
  In mathematics, specifically abstract algebra, the isomorphism theorems are three theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures...
- Fundamental theorem on homomorphisms
  Fundamental theorem on homomorphisms
  In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the kernel and image of the homomorphism....
Universal property
Universal property
In various branches of mathematics, a useful construction is often viewed as the “most efficient solution” to a certain problem. The definition of a universal property uses the language of category theory to make this notion precise and to study it abstractly.This article gives a general treatment...
Filtration (mathematics)
Category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
- Monoidal category
  Monoidal category
  In 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...
- Groupoid
  Groupoid
  In mathematics, especially in category theory and homotopy theory, a groupoid generalises the notion of group in several equivalent ways. A groupoid can be seen as a:...
- Group object
  Group object
  In category theory, a branch of mathematics, group objects are certain generalizations of groups which are built on more complicated structures than sets...
- Coalgebra
  Coalgebra
  In mathematics, coalgebras or cogebras are structures that are dual to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams...
- Bialgebra
  Bialgebra
  In mathematics, a bialgebra over a field K is a vector space over K which is both a unital associative algebra and a coalgebra, such that these structures are compatible....
- Hopf algebra
  Hopf algebra
  In 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...
- Magma object
Torsion (algebra)

Computer algebra

Symbolic mathematics
- Finite field arithmetic
  Finite field arithmetic
  Arithmetic in a finite field is different from standard integer arithmetic. There are a limited number of elements in the finite field; all operations performed in the finite field result in an element within that field....
- Gröbner basis
  Gröbner basis
  In computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating subset of an ideal I in a polynomial ring R...
- Buchberger's algorithm
  Buchberger's algorithm
  In computational algebraic geometry and computational commutative algebra, Buchberger's algorithm is a method of transforming a given set of generators for a polynomial ideal into a Gröbner basis with respect to some monomial order. It was invented by Austrian mathematician Bruno Buchberger...