# Endomorphism ring

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In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object; this may be denoted End(X). As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.

### Examples

The elements of the endomorphism ring of an abelian group (A, +) are the endomorphisms of A, i.e. the group homomorphisms 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 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 is the identity map 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.[1]

We can define the endomorphism ring of any module 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 and we consider the K-vector space Kn, then the endomorphism ring of Kn (which consists of all K-linear maps from Kn to Kn) is naturally identified with the ring of n-by-n matrices with entries in K.[2] More generally, the endomorphism algebra of the free module M = Rn is naturally n-by-n matrices with entries in R.

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

### Properties

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