Amalgamation of rings defined by bézout-like conditions

Abstract

Let $f:A\lo B$ be a ring homomorphism and let $J$ be an ideal of $B.$ In this paper, we investigate the transfer of notions elementary divisor ring, Hermite ring and Bézout ring to the amalgamation $A{\bowtie }^{f}J.$ We provide necessary and sufficient conditions for $A{\bowtie }^{f}J$ to be an elementary divisor ring where $A$ and $B$ are integral domains. In this case it is shown that $A{\bowtie }^{f}J$ is an Hermite ring if and only it is a Bézout ring. In particular, we study the transfer of the previous notions to the amalgamated duplication of a ring $A$ along an $A-$submodule $E$ of $Q(A)$ such that $E^2\subseteq E.$