Amalgamation of rings defined by b\'ezout-like conditions

TL;DR

This paper explores how properties like Bézout, Hermite, and elementary divisor rings transfer to amalgamated rings formed via ring homomorphisms, providing conditions for these properties in integral domain contexts.

Contribution

It establishes necessary and sufficient conditions for the transfer of ring properties to amalgamated rings and studies their equivalence in specific constructions like amalgamated duplication.

Findings

A necessary and sufficient condition for $A\bowtie^fJ$ to be an elementary divisor ring when $A$ and $B$ are integral domains.

In this setting, $A\bowtie^fJ$ is Hermite if and only if it is a Bézout ring.

Application to the transfer of properties in the amalgamated duplication of a ring along a submodule.

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 B\'ezout ring to the amalgamation $A\bowtie^fJ.$ We provide necessary and sufficient conditions for $ A\bowtie^fJ$ to be an elementary divisor ring where $A$ and $B$ are integral domains. In this case it is shown that $ A\bowtie^fJ$ is an Hermite ring if and only it is a B\'ezout 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 $E^2\subseteq E.$