I just learned some basic notion on localization of a category, and I'm curious about its linkage to the usage localization of a module.
Obviously, any ring R (not necessarily commutative) can be thought as the preadditive category of single object (still call it R), and localization at a left (resp, right) multiplicative system S, actually yields the usual ring of left (resp, right) fractions of R at the corresponding S.
I'm following notations in https://stacks.math.columbia.edu/tag/04VB
See also https://en.wikipedia.org/wiki/Ore_condition
So my question proceeds: can we interpret an R-module M as a category in the sense that, the localization of this category at a multiplicative system S will give the usual localization of M?
Or more loosely, can we see the R-module M as a subcategory of some category, such that the localization of this bigger category at a multiplicative system S will contain the usual localization of M as a subcategory?
The main point is that we need to find a category where morphisms contain arrows that represent elements in M and R. Any suggestion or hint is appreciated. Thanks!