Trivial extensions defined by Prufer conditions

TL;DR

This paper explores how Prufer conditions transfer to trivial ring extensions, generating new examples of rings with zero divisors, and provides evidence supporting conjectures on Gaussian rings and polynomials.

Contribution

It introduces new families of rings with Prufer properties via trivial extensions, expanding understanding of these conditions in rings with zero divisors.

Findings

Generated new examples of Prufer rings with zero divisors.

Provided evidence supporting the Bazzoni-Glaz conjecture.

Extended the scope of the Kaplansky-Tsang conjecture.

Abstract

This paper deals with well-known extensions of the Prufer domain concept to arbitrary commutative rings. We investigate the transfer of these notions in trivial ring extensions (also called idealizations) of commutative rings by modules and then generate original families of rings with zerodivisors subject to various Prufer conditions. The new examples give further evidence for the validity of Bazzoni-Glaz conjecture on the weak dimension of Gaussian rings. Moreover, trivial ring extensions allow us to widen the scope of validity of Kaplansky-Tsang conjecture on the content ideal of Gaussian polynomials.