In abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object.

We will start with the example of abelian groups. Suppose A is an abelian group. As the name suggests, the elements of the endomorphism ring of 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 (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. Then 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 group homomorphisms.

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]

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

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

• If a module is simple, then its endomorphism ring is a division ring (this is sometimes called Schur's lemma). ^[3]
• A module is indecomposable if and only if its endomorphism ring does not contain any non-trivial idempotents. ^[4]

References