modules over algebars and homomorphisms between them
Let $R$ be a ring and let $A$ be an associative algebra (not necessarily unital).
Definition. A (left) $A$module over $R$ is a pair $(M,\circ )$ where $M$ is a (left) $R$module and
$$\circ :A\times M\to M$$ 
is a $R$bilinear map such that the following conditions hold:

1.
$(a\circ b)\circ x=a\circ (b\circ x)$

2.
$r(a\circ x)=(ra)\circ x=a\circ (rx)$
for any $a,b\in A$, $x\in M$ and $r\in R$. We will simply use capital letters to denote modules.
Let $M$ be an $A$module over $R$. If ${M}^{\prime}\subseteq M$ and ${A}^{\prime}\subseteq A$ then by ${A}^{\prime}{M}^{\prime}$ we denote $R$submodule^{} of $M$ generated by elements of the form $am$ for $a\in {A}^{\prime}$ and $m\in {M}^{\prime}$. We will call $M$ unitary if $AM=M$. Note, that if $A$ has multiplicative identity^{} $1$, then $M$ is unitary if and only if $1m=m$ for any $m\in M$.
The reason we use name ,,$A$module over $R$” instead of ,,$A$module” is that these to concepts may differ. The latter means that we treat $A$ simply as a ring and take modules over it. But such module need not be equiped with a ,,good” $R$module structure^{}. On the other hand this is always the case, when $M$ is unitary over unital algebra^{}.
If $M$ and $N$ are two $A$modules over $R$, then a function $f:M\to N$ is called an $A$homomorphism^{} iff $f$ is an $R$homomorphism and additionaly $f(am)=af(m)$ for any $a\in A$ and $m\in M$.
It can be easily checked that $A$modules over $R$ together with $A$homomorphisms form a category^{} which is abelian^{}. Furthermore, if $A$ is unital, then its full subcategory consisting unitary $R$modules over $A$ is equivalent^{} to category of unitary $A$modules.
In most cases it is important to assume that the base ring $R$ is a field, even algebraically closed^{}.
Title  modules over algebars and homomorphisms between them 

Canonical name  ModulesOverAlgebarsAndHomomorphismsBetweenThem 
Date of creation  20130322 19:16:32 
Last modified on  20130322 19:16:32 
Owner  joking (16130) 
Last modified by  joking (16130) 
Numerical id  5 
Author  joking (16130) 
Entry type  Definition 
Classification  msc 13B99 
Classification  msc 20C99 
Classification  msc 16S99 