On Armendariz-like properties in Amalgamated algebras along ideals

TL;DR

This paper explores how Armendariz-like properties transfer to amalgamated algebras along ideals, providing necessary and sufficient conditions for these properties in the algebraic construction.

Contribution

It establishes necessary and sufficient conditions for the transfer of Armendariz-like properties to amalgamated algebras along ideals.

Findings

Characterizes when $A\bowtie^fJ$ is Armendariz

Identifies conditions for nil-Armendariz property transfer

Determines criteria for weak Armendariz property

Abstract

Let $f: A\rightarrow B$ be a ring homomorphism and $J$ be an ideal of $B$. In this paper, we investigate the transfer of Armendariz-like properties to the amalgamation of $A$ with $B$ along $J$ with respect to $f$ (denoted by $A\bowtie^fJ)$ introduced and studied by D'Anna, Finocchiaro and Fontana in 2009. Our aim is to provide necessary and sufficient conditions for $A\bowtie^fJ,$ to be an Armendariz ring, nil-Armendariz ring and weak Armendariz ring.