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http://hdl.handle.net/20.500.11889/7763
Title: | On locally divided rings and going-down rings | Authors: | Badawi, Ayman Dobbs, David E. |
Keywords: | Commutative rings | Issue Date: | 2001 | Publisher: | Communications in Algebra | 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. | URI: | http://hdl.handle.net/20.500.11889/7763 | DOI: | 10.1081/AGB-4988 |
Appears in Collections: | Fulltext Publications |
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