When every principal ideal is flat

TL;DR

This paper characterizes $PF$-rings where all principal ideals are flat, providing new criteria and conditions for related constructions like ring extensions and quotients.

Contribution

It offers a novel characterization of $PF$-rings and establishes necessary and sufficient conditions for their preservation under certain ring constructions.

Findings

New characterization of $PF$-rings.

Conditions for $R\bowtie I$ and $R/I$ to be $PF$-rings.

Discussion on the scope and limitations of the results.

Abstract

This paper deals with well-known notion of $PF$-rings, that is, rings in which principal ideals are flat. We give a new characterization of $PF$-rings. Also, we provide a necessary and sufficient condition for $R\bowtie I$ (resp., $R/I$ when $R$ is a Dedekind domain or $I$ is a primary ideal) to be $PF$-ring. The article includes a brief discussion of the scope and precision of our results.