On $\phi$-Pr\"ufer like conditions

TL;DR

This paper explores conditions under which $\phi$-rings are $\phi$-Pr"ufer, using lattice and content ideal techniques, and establishes key properties and classifications of these rings.

Contribution

It introduces new criteria for $\phi$-Pr"ufer rings, demonstrating that Gaussian $\phi$-rings are $\phi$-Pr"ufer and characterizing semi-local $\phi$-Pr"ufer rings as $\phi$-Bézout.

Findings

Every Gaussian $\phi$-ring is $\phi$-Pr"ufer.

Semi-local $\phi$-Pr"ufer rings are $\phi$-Bézout.

Analysis of lattice structure and content ideals in $\phi$-rings.

Abstract

In this paper, we investigate the question of when a $\phi$-ring is $\phi$-Pr\"ufer using two types of techniques: first, by analysing the lattice structure of the nonnil ideals of $\phi$-rings; and secondly, by considering content ideal techniques which were developed to study Gaussian polynomials. In particular, we conclude that every Gaussian $\phi$-ring is $\phi$-Pr\"ufer. Key concepts such as $\phi$-weak global dimension, primary ideals and irreducible ideals are discussed, along with their hereditary properties in $\phi$-Pr\"ufer rings. We also prove that any semi-local $\phi$-Pr\"ufer ring is a $\phi$-B\'ezout ring. This paper includes several theorems and examples that provide insights into the $\phi$-Pr\"ufer rings and their implications in the field of ring theory.