On locally divided rings and going-down rings

Abstract

All rings considered below are commutative with 1. As in [B2], if R is a ring, P 2 SpecðRÞ is a divided prime ideal in R if P is comparable under inclusion to each ideal of R; and R is a divided ring if each P 2 SpecðRÞ is divided in R. Divided rings generalize the divided domains introduced in [D2]. Our main goal is to generalize another class of domains introduced in [D2], the locally divided domains. We say that a ring R is a locally divided ring if RP is a divided ring for each P 2 SpecðRÞ. Each divided ring is locally divided [B2, Proposition 4]. Since the literature on locally divided domains ([D2], [D4], [DF]) is tied to studies of going-down domains (in the sense of [D1]), it is natural to pursue connections between locally divided rings and the recently introduced going-down rings [D5]. Section 3 develops several such connections, some with the flavor of domain-theoretic studies and others differing from such phenomena in the presence of zero-divisors. First, Section 2 develops more information about divided rings and initiates the theory of locally divided rings.