Let be a ring. A left -module is an abelian group together with a function , denoted by juxtaposition, that satisfies the following axioms for all and :
Let , be left modules over a ring . A function is called homogenous if and only if for all and the identity
Let , be left modules over a ring . A function is called linear if and only if it is both homogenous and a morphism of abelian groups from to .
Theorem (first isomorphism theorem):
Let and be left modules over a ring . Let be linear. Then