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## Summary

In mathematics, the endomorphisms of an abelian group X form a ring. This ring is called the endomorphism ring X, denoted by End(X); the set of all homomorphisms of X into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map ${\textstyle 0:x\mapsto 0}$ as additive identity and the identity map ${\textstyle 1:x\mapsto x}$ as multiplicative identity.

The functions involved are restricted to what is defined as a homomorphism in the context, which depends upon the category of the object under consideration. The endomorphism ring consequently encodes several internal properties of the object. As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.

An abelian group is the same thing as a module over the ring of integers, which is the initial ring. In a similar fashion, if R is any commutative ring, the endomorphism monoids of its modules form algebras over R by the same axioms and derivation. In particular, if R is a field F, its modules M are vector spaces V and their endomorphism rings are algebras over the field F.

## Description

Let (A, +) be an abelian group and we consider the group homomorphisms from A into A. Then addition of two such homomorphisms may be defined pointwise to produce another group homomorphism. Explicitly, given two such homomorphisms f and g, the sum of f and g is the homomorphism ${\textstyle [f+g](x):=f(x)+g(x)}$ . Under this operation End(A) is an abelian group. With the additional operation of composition of homomorphisms, End(A) is a ring with multiplicative identity. This composition is explicitly ${\textstyle (fg)(x):=f(g(x))}$ . The multiplicative identity is the identity homomorphism on A.

If the set A does not form an abelian group, then the above construction is not necessarily additive, as then the sum of two homomorphisms need not be a homomorphism. This set of endomorphisms is a canonical example of a near-ring that is not a ring.

## Examples

• In the category of R modules the endomorphism ring of an R-module M will only use the R module homomorphisms, which are typically a proper subset of the abelian group homomorphisms. When M is a finitely generated projective module, the endomorphism ring is central to Morita equivalence of module categories.
• For any abelian group $A$ , $M_{n}(\operatorname {End} (A))\cong \operatorname {End} (A^{n})$ , since any matrix in $M_{n}(\operatorname {End} (A))$  carries a natural homomorphism structure of $A^{n}$  as follows:
${\begin{pmatrix}\varphi _{11}&\cdots &\varphi _{1n}\\\vdots &&\vdots \\\varphi _{n1}&\cdots &\varphi _{nn}\end{pmatrix}}{\begin{pmatrix}a_{1}\\\vdots \\a_{n}\end{pmatrix}}={\begin{pmatrix}\sum _{i=1}^{n}\varphi _{1i}(a_{i})\\\vdots \\\sum _{i=1}^{n}\varphi _{ni}(a_{i})\end{pmatrix}}.$
One can use this isomorphism to construct a lot of non-commutative endomorphism rings. For example: $\operatorname {End} (\mathbb {Z} \times \mathbb {Z} )\cong M_{2}(\mathbb {Z} )$ , since $\operatorname {End} (\mathbb {Z} )\cong \mathbb {Z}$ .
Also, when $R=K$  is a field, there is a canonical isomorphism $\operatorname {End} (K)\cong K$ , so $\operatorname {End} (K^{n})\cong M_{n}(K)$ , that is, the endomorphism ring of a $K$ -vector space is identified with the ring of n-by-n matrices with entries in $K$ . More generally, the endomorphism algebra of the free module $M=R^{n}$  is naturally $n$ -by-$n$  matrices with entries in the ring $R$ .
• As a particular example of the last point, for any ring R with unity, End(RR) = R, where the elements of R act on R by left multiplication.
• In general, endomorphism rings can be defined for the objects of any preadditive category.