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Let $f:A\to B$ and $g:A\to C$ be two ring homomorphisms and let $J$ and $J^{\text{'}}$ be ideals of $B$ and $C$, respectively, such that $f^{-1}\left(J\right)=g^{-1}\left(J^{\text{'}}\right)$. In this paper, we investigate the transfer of the notions of Gaussian and Prüfer rings to the bi-amalgamation of $A$ with $(B,C)$ along $(J,J^{\text{'}})$ with respect to $(f,g)$ (denoted by $A{\bowtie }^{f,g}(J,J^{\text{'}})),$ introduced and studied by S. Kabbaj, K. Louartiti and M. Tamekkante in 2013. Our results recover well known results on amalgamations in C. A. Finocchiaro (2014) and generate new original examples of rings possessing these properties.

This paper aims at the study of the notions of periodic, UU and semiclean properties in various context of commutative rings such as trivial ring extensions, amalgamations and pullbacks. The results obtained provide new original classes of rings subject to various ring theoretic properties.