Most older priests I know would wear a ring, both as a symbol to their union with the Church, and as a means of warding off any potential female while not in their clerics. I don’t believe it is a requirement though. I asked my parish priest once why he didn’t wear one and he jokingly replied that he hasn’t met the right girl yet,
I cannot speak for religious priests (Br JR please chime in here if you wish), but secular priests (with the possible exception of those who are married under the “Pastoral Provision” but even then only the wedding band would come into play) should not wear rings while celebrating Mass. Even Canon Lawyers are supposed to remove their rings before Mass.
I’m not sure why this should create confusion. Canon Lawyers are given a ring (which was traditionally black onyx), as well as a biretta with a green pom, as a sign of academic achievement. Neither the ring or the biretta with green pom is worn at Mass. Ditto the black zuchetto. OTOH, bishops are entitled to a ring (traditionally amethyst, but that seems to have have changed in post-conciliar times) as a symbol of the episcopacy (i.e. the fullness of the priesthood) itself. Simple (celibate and secular) priests were never given a ring. In any case, by tradition, only bishops wear their ring while celebrating Mass.
One could always ask a traditionally educated bishop or perhaps a similarly educated canon lawyer.