# Linear Algebra over a Ring/Modules and linear functions

Definition (module):

Let ${\displaystyle R}$ be a ring. A left ${\displaystyle R}$-module is an abelian group ${\displaystyle (M,+)}$ together with a function ${\displaystyle R\times M\to M}$, denoted by juxtaposition, that satisfies the following axioms for all ${\displaystyle r,s\in R}$ and ${\displaystyle m,n\in M}$:

1. ${\displaystyle 1m=m}$
2. ${\displaystyle (r+s)m=rm+sm}$
3. ${\displaystyle r(sm)=(rs)m}$
4. ${\displaystyle r(m+n)=rm+rn}$

Definition (homogenous):

Let ${\displaystyle M}$, ${\displaystyle N}$ be left modules over a ring ${\displaystyle R}$. A function ${\displaystyle f:M\to N}$ is called homogenous if and only if for all ${\displaystyle r\in R}$ and ${\displaystyle m\in N}$ the identity

${\displaystyle f(rm)=rf(m)}$

holds.

Definition (linear):

Let ${\displaystyle M}$, ${\displaystyle N}$ be left modules over a ring ${\displaystyle R}$. A function ${\displaystyle f:M\to N}$ is called linear if and only if it is both homogenous and a morphism of abelian groups from ${\displaystyle M}$ to ${\displaystyle N}$.

Theorem (first isomorphism theorem):

Let ${\displaystyle M}$ and ${\displaystyle N}$ be left modules over a ring ${\displaystyle R}$. Let ${\displaystyle \varphi :M\to N}$ be linear. Then

${\displaystyle M/\ker \varphi \cong \operatorname {im} \varphi }$.

Proof: ${\displaystyle \Box }$

## Exercises

1. Prove that for a function ${\displaystyle f:M\to N}$  between left ${\displaystyle R}$ -modules, the following are equivalent:
1. ${\displaystyle f}$  is linear
2. For all ${\displaystyle m,n\in M}$  and ${\displaystyle r\in R}$ , we have ${\displaystyle f(m+n)=f(m)+f(n)}$  and ${\displaystyle f(rm)=rf(m)}$
3. For all ${\displaystyle m,n\in M}$  and ${\displaystyle r\in R}$ , we have ${\displaystyle f(m+rn)=f(m)+rf(n)}$
4. For all ${\displaystyle m,n\in M}$  and ${\displaystyle r,s\in R}$ , we have ${\displaystyle f(rm+sn)=rf(m)+sf(n)}$