Local types of Prüfer rings

Abstract

Earlier papers by Glaz, Boynton, and the authors established hierarchies of properties for commutative rings in general which become equivalent to “Prüfer domain” for integral domains. Between Gaussian rings and Prüfer rings, these papers considered locally (respectively, maximally) Prüfer (respectively, strong Prüfer) rings. In this paper, we refine these hierarchies still further by considering the restriction of these local conditions to just the regular (respectively, semiregular) prime (respectively, maximal) ideals of the ring. In addition, we also consider these local conditions for [Formula: see text]-Prüfer rings. This refinement leads to a hierarchy of 21 conditions between locally strong Prüfer rings and Prüfer rings, of which 17 are inequivalent. We also determine the extent to which these conditions pass between a ring and its total quotient ring.