I have no idea if this is trivial or not, though I've been searching on internet quite a bit and still seems to be pretty vague in my head and didn't get a real answer.
In a module category over a ring $R$, we know that no matter what our direct/inverse system is, we can form its associated direct/inverse limit, which turns out to have an explicit description more or less given in any standard textbook of abstract algebra nowadays. However, I haven't figured out if the above is still true in case of (commutative) rings (perhaps an answer in the non-commutative setting if does exist would be appreciated too). Can you help me out please?
Commutative and non-commutative rings are models of a Lawvere theory so the categories of them are complete and cocomplete (i.e. have all limits [inverse limits] and colimits [direct limits]). Indeed, most things you think of as categories of "algebraic" structures are categories of models of a Lawvere theory and thus are complete and cocomplete. There are some exceptions, e.g. fields and integral domains are not models of a Lawvere theory. (To be more precise, the category of fields is not a category of models of a Lawvere theory. Obviously, fields are rings and so are models of the Lawvere theory of rings.)