Gaussian and Pr\"ufer conditions in bi-amalgamated algebras

TL;DR

This paper explores how Gaussian and Pr"ufer properties transfer to bi-amalgamated rings formed from two ring homomorphisms and ideals, extending previous results and providing new examples.

Contribution

It investigates the transfer of Gaussian and Pr"ufer properties to bi-amalgamated rings, generalizing known results and introducing new examples.

Findings

Recovered known results on amalgamations.

Established transfer conditions for Gaussian and Pr"ufer properties.

Generated new examples of rings with these properties.

Abstract

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two ring homomorphisms and let $J$ (resp., $J'$) be an ideal of $B$ (resp., $C$) such that $f^{-1}(J)=g^{-1}(J')$. In this paper, we investigate the transfer of the notions of Gaussian and Pr\"ufer properties to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J')$ with respect to $(f,g)$ (denoted by $A\bowtie^{f,g}(J,J')),$ introduced and studied by Kabbaj, Louartiti and Tamekkante in 2013. Our results recover well known results on amalgamations in \cite{Finno} and generate new original examples of rings satisfying these properties.