ℎ-local Rings

In the 1960’s, Matlis defined an h h -local domain to be a (commutative) integral domain in which each nonzero element is contained in only finitely many maximal ideals and each nonzero prime ideal is contained in a unique maximal ideal. For rings with zero-divisors, by changing “nonzero” to “regular,” one obtains the definition of an h h -local ring. Nearly two dozen equivalent characterizations of h h -local domain have appeared in the literature. We show that most of these remain equivalent to h h -local ring if one also replaces “localization” by “regular localization” and assumes that the ring is a Marot ring (i.e., every regular ideal is generated by its regular elements).