Bi-amalgamated algebras along ideals

Abstract

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms, and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \textit {bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by \[ A\bowtie ^{f,g}(J,J'):=\big \{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big \}. \] In this paper, we investigate ring-theoretic properties of \textit {bi-amalgamations} and capitalize on previous work carried out on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations, and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna, Finocchiaro and Fontana's amalgamations and duplications.