Commutative FPF Rings Arising as Split-Null Extensions

Abstract

Let $R = (B,E)$ be the split-null or trivial extension of a faithful module $E$ over a commutative ring $B$. $R$ is an FPF ring iff the partial quotient ring $B{S^{ - 1}}$ with respect to the set $S$ of elements of $B$ with zero annihilator in $E$ is canonically the endomorphism ring of $E$, that is $B{S^{ - 1}} = {\operatorname {End}_B}E{S^{ - 1}}$, every finitely generated ideal with zero annihilator in $E$ is invertible in $B{S^{ - 1}}$, and $E = E{S^{ - 1}}$ is an injective module over $B$. The proof uses the author’s characterization of commutative FPF rings [1] and also the characterization of self-injectivity of a split-null extension [3].