Linear Algebra over a Ring/Modules and linear functions

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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:

Exercises[edit | edit source]

  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