It seems that there is no clear consensus about tagging questions about (or related to) rings.
There are several tags which might apply.
This tag is for questions about rings, which are a type of algebraic structure studied in abstract algebra and algebraic number theory.
Questions about commutative rings, their ideals, and their modules.
For questions about rings which are not necessarily commutative and modules over such rings.
Personally if a question was related to rings I usually add the ring-theory tag (if it is missing). (To clarify: I do not claim that whenever the word "ring" is mentioned in a post, then it should have this tag. But if rings are main topic or very relevant to the question, then I tend to add the tag rather than just leaving some rather general umbrella tag such as abstract-algebra.) I think that this improves possibility when searching for older questions. I also think that this usage is rather intuitive - use rings for questions about rings, use group for questions about groups.
However, I was told on two occasions by a rather experienced users that it would be better to use abstract-algebra rather than ring-theory or commutative-algebra in the cases where the question is too elementary. (In both cases this occurred in questions about ideal in some specific ring. Such questions would, in my opinion, deserve also some ring-related tag.) At the time when we exchange some comments on the main site about this issue, I have also checked their recent activity and seen that they have indeed done some retags removing ring-theory or commutative-algebra and leaving abstract-algebra. (I will add that this user is definitely very knowledgeable of these areas, they have gold badges both in abstract-algebra and commutative-algebra. And this user is also very active in editing and improving post, including retagging.)
As I do not want to misrepresent their position, I will quote their comments in full:
I think this is a basic abstract algebra question. Usually I prefer to leave the specific questions to fall under the commutative-algebra tag, but if you want you can restore it. (Not every question about commutative rings belong to commutative algebra. For me, that description of the tag is too general and a little bit silly.)
@MartinSleziak I'm not sure that every question dealing with a ring belongs to
ring-theoryas this is usually understood. But on M.SE the confusion with tags is ubiquitous.
Personally, I am not very active in questions from algebra. But if I see a question which is incorrectly tagged (at least in my opinion) I try to help by improving the tags. I agree with the point which Jyrki Lahtonen made in this answer that various subcommunities naturally arise around interconnected tags and that users actives in those subcommunities should have at least some say in how questions in those tags are treated. (That post was made in context of off-topic/mising context closures, but the point he made there probably applies to tagging, too.) As far as the tags I mentioned are concerned, I have no doubt that some users follow them closely and do their best to correct wrong retags. Nevertheless, it would be good to have the official position somewhere officially documented and to have at least some kind of consensus. (If not for other reasons at least to achieve that well meaning users do not make retags which have later to be corrected by other users.)
TL:DR; To summarize what I am asking: