ra.rings and algebras - characterization of a submodule

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}= {($M$,$Rbigotimes _{k}Mrightarrow M$).

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

Let $R_{1}=Rbigoplus k$ 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 $R_{1}-mod$

$R_{1}-modoverset{q_{R}^{*}}{rightarrow}R-mod$ 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

$m-mod$=$A-mod/({Mepsilon A-mod|mcdot M=0})^{-}$, 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

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