First, Let me talk about the "correct definition" of module over non-unital ring(not necessarily commutative) and how this definition coincide with usual definition of module over unital ring in particular case

First we study $R-mod_{1}$={category of associative action of $R$ on $k$-mod}= {($M$,$R\bigotimes _{k}M\rightarrow M$).

$r_{1}(r_{2}z)=(r_{1}r_{2})z$}

Let $R_{1}=R\bigoplus k$ be an untial $k$-algebra with usual multiplication. And we have the categorical equivalence as: $R-mod_{1}\approx R_{1}-mod$

Now,we define module over non-unital algebra $R$ as $R-mod=R_{1}-mod/(Tors_{R_{1}})^{-}$, where $(Tors_{R_{1}})^{-}$ is Serre subcategory of $R_{1}-mod$

$R_{1}-mod\overset{q_{R}^{*}}{\rightarrow}R-mod$ is a localization functor having right adjoint functor.

**Trivial Example:**

if $R$ has is an unital $k$-algebra. Then $R_{1}-mod$ is equivalent to $R-mod$

**Less Trivial example in commutative case:**

Consider affine line $k[x]$. Let $R=xk[x]$(maximai ideal of $k[x]$). Then $R-mod$=$Qcoh(\mathbb{A}^{1}-{0}$). It is a cone.

**Toy general case:**

Let $m$ is a two-sided proper ideal of associative commutative unital ring $A$. Then: we have

$m-mod$=$A-mod/({M\epsilon A-mod|m\cdot M=0})^{-}$, where$T^{-}$ is smallest Serre category containing $T$. It is clear that is equivalent to Qcoh(Complement of $\mathbb{V}(m)$),where
$\mathbb{V}(m)$ is closed subvariety determined by $m$.

Now, I should stop here and write another(maybe)post on definition of sub-module. There are several reference:

Gabriel, Pierre Des catégories abéliennes. (French) Bull. Soc. Math. France 90 1962 323--448
Kontsevich-Rosenberg Noncommutative spaces and flat descent

Gabber-RameroAlmost Ring Theory

justifyany further retagging in the "notes on edit" field. – Andrew Stacey Feb 11 '10 at 10:28