Bi-Amalgamated algebras along ideals

Abstract

Let $f:A\to B$ and $g:A\to C$ be two commutative ring homomorphisms and let $J$ and $J^{\prime }$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J^{\prime })$. The \emph{bi-amalgamation} of $A$ with $(B,C)$ along $(J,J^{\prime })$ with respect to $(f,g)$ is the subring of $B\times C$ given by $$A{\bowtie }^{f,g}(J,J^{\prime }):=\{(f(a)+j,g(a)+j^{\prime })\mid a\in A,(j,j^{\prime })\in J\times J^{\prime }\}.$$ This paper investigates ring-theoretic properties of \emph{bi-amalgamations} and capitalizes on previous works carried 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-Fontana's amalgamations and duplications.

15 pages