# 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

- 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