Some Analogues of a Result of Vasconcelos
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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 (＊) if for each nonzero element a in R, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (＊), 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 (＊) 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 (＊), but the converse is false.
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 (＊) if for each nonzero element a in R, each monomorphic R-module endomorphism of R/Ra is an isomorphism. If R has property (＊), 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 (＊) 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 (＊), but the converse is false.
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