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 ($star$) if for each nonzero element $a in R$, each monomorphic $R$-module endomorphism of $R/Ra$ is an isomorphism. If $R$ has property ($star$), 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 ($star$) 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 ($star$), 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