Given a module M over a unit ring R, the set End_R(M) of its module endomorphisms is a ring with respect to the addition of maps, (f + g)(x) = f(x) + g(x), for all x element M, and the product given by map composition, (f g)(x) = f°g(x) = f(g(x)), for all x element M. The endomorphism ring of M is, in general, noncommutative, but it is always a unit ring (its unit element being the identity map on M).

endomorphism | Schur's lemma