* Given an R-module M, the endomorphism ring of M, denoted End < sub > R </ sub >( M ) is an R-algebra by defining ( r · φ )( x ) = r · φ ( x ).

In general, if R is a ring and S is a simple module over R, then, by Schur's lemma, the endomorphism ring of S is a division ring ; every division ring arises in this fashion from some simple module.

In the case when the endomorphism ring, where each endomorphism arises as left multiplication by a fixed ring element.

If a is an idempotent of the endomorphism ring End < sub > R </ sub >( M ), then the endomorphism is an R module involution of M. That is, f is an R homomorphism such that f < sup > 2 </ sup > is the identity endomorphism of M.

Consequently the endomorphism ring of any simple module is a division ring.

For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q.

For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

Focusing on a single object A in a preadditive category, these facts say that the endomorphism hom-set Hom ( A, A ) is a ring, if we define multiplication in the ring to be composition.

This ring is the endomorphism ring of A. Conversely, every ring ( with identity ) is the endomorphism ring of some object in some preadditive category.