=====Modules over a ring=====

Abbreviation: **RMod**
====Definition====
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 $rx$.

==Morphisms==
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 $f_r$:

$h(f_r(x))=f_r(h(x))$
====Examples====
Example 1: 

====Basic results====


====Properties====
^[[Classtype]]  |variety |
^[[Equational theory]]  | |
^[[Quasiequational theory]]  | |
^[[First-order theory]]  | |
^[[Locally finite]]  |no |
^[[Residual size]]  |unbounded |
^[[Congruence distributive]]  |no |
^[[Congruence modular]]  |yes |
^[[Congruence n-permutable]]  |yes, $n=2$ |
^[[Congruence regular]]  |yes |
^[[Congruence uniform]]  |yes |
^[[Congruence extension property]]  |yes |
^[[Definable principal congruences]]  |no |
^[[Equationally def. pr. cong.]]  |no |
^[[Amalgamation property]]  | |
^[[Strong amalgamation property]]  | |
^[[Epimorphisms are surjective]]  | |
====Finite members====

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

====Subclasses====
====Superclasses====
[[Abelian groups]] 


====References====

[(Ln19xx>
)]