Semi-regular flat modules over strong Pr\"{u}fer rings

TL;DR

This paper introduces semi-regular flat modules and characterizes strong Prüfer rings through properties of these modules, including submodules, ideals, and module envelopes.

Contribution

It defines semi-regular flat modules and establishes their equivalence with strong Prüfer ring properties, providing new module-theoretic characterizations.

Findings

A ring is strong Prüfer iff every submodule of a semi-regular flat module is semi-regular flat.

A ring is strong Prüfer iff every ideal is semi-regular flat.

A ring is strong Prüfer iff every module has a surjective semi-regular flat preenvelope.

Abstract

We first introduce and study the notion of semi-regular flat modules, and then show that a ring $R$ is a strong \Prufer\ ring if and only if every submodule of a semi-regular flat $R$-module is semi-regular flat, if and only if every ideal of $R$ is semi-regular flat, if and only if every $R$-module has a surjective semi-regular flat (pre)envelope.