Editing Module homomorphism (section)

== Terminology ==
A module homomorphism is called a ''module isomorphism'' if it admits an inverse homomorphism; in particular, it is a [[bijection]]. Conversely, one can show a bijective module homomorphism is an isomorphism; i.e., the inverse is a module homomorphism. In particular, a module homomorphism is an isomorphism [[if and only if]] it is an isomorphism between the underlying abelian groups.

The [[isomorphism theorem]]s hold for module homomorphisms.

A module homomorphism from a module ''M'' to itself is called an [[endomorphism]] and an isomorphism from ''M'' to itself an [[automorphism]]. One writes <math>\operatorname{End}_R(M) = \operatorname{Hom}_R(M, M)</math> for the set of all endomorphisms of a module ''M''. It is not only an abelian group but is also a ring with multiplication given by function composition, called the [[endomorphism ring]] of ''M''. The [[group of units]] of this ring is the [[automorphism group]] of ''M''. 

[[Schur's lemma]] says that a homomorphism between [[simple module]]s (modules with no non-trivial [[submodule]]s) must be either zero or an isomorphism. In particular, the endomorphism ring of a simple module is a [[division ring]].

In the language of the [[category theory]], an injective homomorphism is also called a [[monomorphism]] and a surjective homomorphism an [[epimorphism]].