Some Analogues of a Result of Vasconcelos.

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 ∈ 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 FROM AUTHOR]