Not every module has a simple submodule; consider for instance the '''Z'''-module '''Z''' in light of the first example above.
IfLet ''SM'' isand a''N'' simplebe module(left or right) modules over the same ring, and let {{nowrap begin}}''f'' : ''SM'' → ''TN''{{nowrap isend}} be a [[module homomorphism]]. If ''M'' is simple, then ''f'' is either the zero homomorphism or [[injective]]. This is because the [[Kernelkernel (algebra)|kernel]] of ''f'' is a submodule of ''S'' and thus is, by the definition of a simple module, either 0 or ''SM''. If ''TN'' is also a simple module, then ''f'' is either the zero homomorphism or an [[isomorphismsurjective]]. This is because the [[image (mathematics)|image]] of ''f'' is a submodule of ''TN''. andIf thus{{nowrap begin}}''M'' = ''N''{{nowrap end}}, then ''f'' is eitheran 0[[endomorphism]] orof ''TM''., Takenand togetherif ''M'' is simple, thisthen impliesthe prior two statements imply that ''f'' is either the zero homomorphism or an isomorphism. Consequently the [[endomorphism ring]] of any simple module is a [[division ring]]. This result is known as '''[[Schur's lemma]]'''.
The converse of Schur's lemma is not true in general. For example, the '''Z'''-module [[rational number|'''Q''']] is not simple, but its endomorphism ring is isomorphic to the field '''Q'''.
