Some Analogues of a Result of Vasconcelos

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 and only R has Krull dimension 0. Each monomorphic R-module endomorphism of R is an isomorphism and only R = K. We say that R has property () for each nonzero element , 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 () and only 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.