The Stacks project

Proposition 108.73.2. Let $A$ be a ring. Let $Q(A)$ denote its total quotient ring (as in Algebra, Example 10.9.8). Let $\text{Mod}_ A$ denote the category of $A$-modules and $\mathcal{T}$ its Serre subcategory of torsion modules. Let $\text{Mod}_{Q(A)}$ denote the category of $Q(A)$-modules. Then there is a canonical equivalence $\text{Mod}_ A/\mathcal{T} \rightarrow \text{Mod}_{Q(A)}$.

Proof. Follows immediately from applying Proposition 108.73.1 to the multiplicative subset $S = \{ f \in A \mid f \text{ is not a zerodivisor in }A\} $, since a module is a torsion module if and only if all of its elements are each annihilated by some element of $S$. $\square$


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