On bi-amalgamated constructions

Let $f:A⟶B,g:A⟶C$ be ring homomorphisms and let $\mathfrak{b}$ (resp., $\mathfrak{c}$) be an ideal of $B$ (resp., $C$) satisfying $f^{-1}(\mathfrak{b})=g^{-1}(\mathfrak{c})$. Recently Kabbaj, Louartiti and Tamekkante defined and studied the following subring $$A{\bowtie }^{f,g}(\mathfrak{b},\mathfrak{c}):=\{(f(a)+b,g(a)+c)\mid a\in A,b\in \mathfrak{b},c\in \mathfrak{c}\}$$ of $B\times C$, called the bi-amalgamation of $A$ with $(B,C)$ along $(\mathfrak{b},\mathfrak{c})$, with respect to $(f,g)$. This ring construction is a natural generalization of the amalgamated algebras, introduced and studied by D’Anna, Finocchiaro and Fontana. The aim of this paper is to continue the investigation started by Kabbaj, Louartiti and Tamekkante, by providing a deeper insigt on the ideal-theoretic structure of bi-amalgamations.