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

Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ (resp., $J^{\mathrm{\prime }}$) be an ideal of $B$ (resp., $C$) such that $f^{-1}(J)=g^{-1}(J^{\mathrm{\prime }})$. 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 $(J,J^{\mathrm{\prime }})$ with respect to $(f,g)$ (denoted by $A{\bowtie }^{f,g}(J,J^{\mathrm{\prime }})),$ 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.