Linear Algebra over a Ring/Modules and linear functions

Definition (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 :

Definition (homogenous):

Let , be left modules over a ring . A function is called homogenous if and only if for all and the identity

holds.

Definition (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 .

Theorem (first isomorphism theorem):

Let and be left modules over a ring . Let be linear. Then

.

Proof:

ExercisesEdit

  1. Prove that for a function   between left  -modules, the following are equivalent:
    1.   is linear
    2. For all   and  , we have   and  
    3. For all   and  , we have  
    4. For all   and  , we have