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

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.