In 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...

, one associates to certain objects a ring

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...

, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End(X). As the resulting object is often an algebra

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....

 over some ring R, this may also be called the endomorphism algebra.

Examples

The elements of the endomorphism ring of an 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...

  are the endomorphism

Endomorphism

In mathematics, an endomorphism is a morphism from a mathematical object to itself. For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about...

s of A, i.e. the group homomorphism

Group homomorphism

In mathematics, given two groups and , a group homomorphism from to is a function h : G → H such that for all u and v in G it holds that h = h \cdot h...

s from A to A. Any two such endomorphisms f and g can be added pointwise (using the formula (f+g)(x) = f(x) + g(x)), and the result f+g is again an endomorphism of A. Furthermore, f and g can also be composed to yield the endomorphism f o g, and this multiplication distributes

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:...

 over pointwise addition. The set of all endomorphisms of A, together with this addition and multiplication, satisfies all the axioms of a ring. This is the endomorphism ring of A. Its multiplicative identity

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...

 is the identity map

Identity function

In mathematics, an identity function, also called identity map or identity transformation, is a function that always returns the same value that was used as its argument...

 on A. Endomorphism rings are typically non-commutative.

The above construction does not work for groups that are not abelian: the sum of two homomorphisms need not be a homomorphism in that case.

We can define the endomorphism ring of any module

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...

 in exactly the same way, using module homomorphisms instead of abelian group homomorphisms; abelian groups are exactly modules over the integers. The result is an algebra over the ring R of scalar transformations.

If K is a field

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...

 and we consider the K-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...

 Kⁿ, then the endomorphism ring of Kⁿ (which consists of all K-linear maps

Linear transformation

In mathematics, a linear map, linear mapping, linear transformation, or linear operator is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. As a result, it always maps straight lines to straight lines or 0...

 from Kⁿ to Kⁿ) is naturally identified with the ring of n-by-n matrices

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...

 with entries in K. More generally, the endomorphism algebra of the 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:...

  is naturally n-by-n matrices with entries in R.

In general, endomorphism rings can be defined for the objects of any preadditive category

Preadditive category

In mathematics, specifically in category theory, a preadditive category is a category that is enriched over the monoidal category of abelian groups...

.

Properties

One can often translate properties of an object into properties of its endomorphism ring. For instance:

• If a module is simple
  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...
  
  , then its endomorphism ring is a 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...
  
   (this is sometimes called 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...
  
  ).
• A module is indecomposable
  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...
  
   if and only if its endomorphism ring does not contain any non-trivial idempotents.