Commutative FPF rings arising as split-null extensions

Abstract

Let R = ( B , E ) R = (B,E) be the split-null or trivial extension of a faithful module E E over a commutative ring B B . R R is an FPF ring iff the partial quotient ring B S − 1 B{S^{ - 1}} with respect to the set S S of elements of B B with zero annihilator in E E is canonically the endomorphism ring of E E , that is B S − 1 = End B E S − 1 B{S^{ - 1}} = {\operatorname {End}_B}E{S^{ - 1}} , every finitely generated ideal with zero annihilator in E E is invertible in B S − 1 B{S^{ - 1}} , and E = E S − 1 E = E{S^{ - 1}} is an injective module over B 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].