Some Analogues of a Result of Vasconcelos

• Journal title : Kyungpook mathematical journal
• Volume 55, Issue 4,  2015, pp.817-826
• Publisher : Department of Mathematics, Kyungpook National University
• DOI : 10.5666/KMJ.2015.55.4.817
Title & Authors
Some Analogues of a Result of Vasconcelos
DOBBS, DAVID EARL; SHAPIRO, JAY ALLEN;

Abstract
Let R be a commutative ring with total quotient ring K. Each monomorphic R-module endomorphism of a cyclic R-module is an isomorphism if and only if R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism if and only if R = K. We say that R has property ($\small{{\star}}$) if for each nonzero element $\small{a{\in}R}$, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property ($\small{{\star}}$), then each nonzero principal prime ideal of R is a maximal ideal, but the converse is false, even for integral domains of Krull dimension 2. An integral domain R has property ($\small{{\star}}$) if and only if R has no R-sequence of length 2; the "if" assertion fails in general for non-domain rings R. Each treed domain has property ($\small{{\star}}$), but the converse is false.
Keywords
Commutative ring;cyclic module;monomorphism;Krull dimension;monoid ring;integral domain;pullback;treed domain;pseudo-valuation domain;total quotient ring;localization;
Language
English
Cited by
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