# Faithfulness of a module: is it a categorical property?

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.

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 $$I$$.)
• 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. Commented Jun 24 at 4:05