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Let $R$ be a (unital) ring and let $M$ be a left $R$-module. The $R$-module $M$ is called faithful if $$rM = 0\implies r = 0$$ holds for every $r\in R$, i.e. the left ideal $$\operatorname{Ann}_R(M)=\{r\in R: rM= 0\}$$ is trivial.

Perhaps a somewhat vague question: Can we characterize the notion of faithfulness of an $R$-module in categorical language? I am looking for a characterization of faithfulness of an $R$-module using a formulation that only involves morphisms and objects in the category of left $R$-modules.

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Here is one such characterization. Say a full subcategory of $R\text{-Mod}$ is good if it is closed under subobjects, quotient objects, and products. Then I claim a module $M$ is faithful iff the good subcategory generated by $M$ is all of $R\text{-Mod}$. In one direction, if some $r\in R$ annihilates $M$, then it annihilates the entire good subcategory generated by $M$, so if $r\neq 0$ then that good subcategory is not all of $R\text{-Mod}$. Conversely, suppose $M$ is faithful and let $C$ be the good subcategory it generates. Then the identity map $M\to M$, as an element of the product module $M^M$, is not annihilated by any nonzero element of $R$, so it generates a submodule of $M^M$ isomorphic to $R$. Thus $R\in C$, and thus every free module is in $C$ (as a submodule of a product of copies of $R$), and thus every module is in $C$ (as a quotient of a free module).

(More generally, a similar argument shows that the good subcategories of $R\text{-Mod}$ are in bijection with two-sided ideals of $R$, with an ideal $I$ corresponding to the subcategory of modules that are annihilated by $I$.)

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  • $\begingroup$ Using the submodule generated by the identity map is very nice! $\endgroup$ Commented Jun 24 at 3:41
  • $\begingroup$ Well, really the idea is just that if you tuple together all the elements of $M$ the annihilator of that tuple is $0$. Describing it as the identity map is just a cute way to do that. $\endgroup$ Commented Jun 24 at 4:05
  • $\begingroup$ That fits nicely with the property of being sincere, meaning every simple module occurs as a composition factor. Then a module is sincere if and only if the Serre subcategory it generates is everything. $\endgroup$ Commented Jun 24 at 5:46

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