Linear Algebra over a Ring/Modules and linear functions

module 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 : homogenous Let , be left modules over a ring . A function is called homogenous if and only if for all and the identity holds. linear 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 . first isomorphism theorem Let and be left modules over a ring . Let be linear. Then .   Exercises Prove that for a function between left -modules, the following are equivalent: is linear For all and , we have and For all and , we have For all and , we have