On the prime spectrum of the amalgamations

TL;DR

This paper investigates the prime spectrum of amalgamated rings formed via a ring homomorphism and ideal, providing characterizations for when these rings are pm-rings and exploring property transfers.

Contribution

It offers new characterizations of the prime spectrum of amalgamated rings and analyzes conditions for pm-ring structures and property transfers.

Findings

Characterization of prime spectrum of amalgamated rings

Conditions for amalgamated rings to be pm-rings

Analysis of property transfer in amalgamated constructions

Abstract

Let $f:R\to S$ be a ring homomorphism and $J$ be an ideal of $S$. Then the subring $R\bowtie^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of $R$ with $S$ along $J$ with respect to $f$. In this paper, we will deepen the study of the prime spectrum of the amalgamations and characterize when $R\bowtie^fJ$ is a pm-ring. We also investigate the transfer of compactly packed and properly zipped property on amalgamated constructions.