(Generalized) filter properties of the amalgamated algebra

TL;DR

This paper investigates the conditions under which the amalgamated algebra formed from two rings and an ideal exhibits filter ring properties, expanding understanding of algebraic structures in ring theory.

Contribution

It characterizes when the amalgamated algebra $R\bowtie^fJ$ is a (generalized) filter ring, providing new insights into the structure of these algebraic constructions.

Findings

Identifies conditions for $R\bowtie^fJ$ to be a filter ring.

Extends known results to generalized filter rings.

Clarifies the structure of amalgamated algebras in relation to filter properties.

Abstract

Let $R$ and $S$ be commutative rings with unity, $f:R\to S$ a ring homomorphism and $J$ an ideal of $S$. Then the subring $R\bowtie^fJ:=\{(a,f(a)+j)\mid a\in R$ 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 determine when $R\bowtie^fJ$ is a (generalized) filter ring.