Abbreviation: RMod

A \emph{module over a rings with identity} $\mathbf{R}$ is a structure $\mathbf{A}=\langle A,+,-,0,f_r\ (r\in R)\rangle$ such that

$\langle A,+,-,0\rangle $ is an abelian groups

$f_r$ preserves addition: $f_r(x+y)=f_r(x)+f_r(y)$

$f_{1}$ is the identity map: $f_{1}(x)=x$

$f_{r+s}(x))=f_r(x)+f_s(x)$

$f_{r\circ s}(x)=f_r(f_s(x))$

Remark: $f_r$ is called \emph{scalar multiplication by $r$}, and $f_r(x)$ is usually written simply as $rx$.

Let $\mathbf{A}$ and $\mathbf{B}$ be modules over a ring $\mathbf{R}$. A morphism from $\mathbf{A}$ to $\mathbf{B}$ is a function $h:A\rightarrow B$ that is a group homomorphism and preserves all $f_r$:

$h(f_r(x))=f_r(h(x))$

Example 1:

$\begin{array}{lr} f(1)= &1
f(2)= &
f(3)= &
f(4)= &
f(5)= &
f(6)= &
\end{array}$

Abelian groups