Minimal Prime Ideals and Semistar Operations

TL;DR

This paper investigates conditions under which the set of minimal prime ideals over a quasi-$igstar$-ideal in a commutative integral domain is finite, focusing on the role of semistar operations of finite type.

Contribution

It establishes that if each minimal prime over a quasi-$igstar$-ideal is the radical of a $igstar$-finite ideal, then the set of such minimal primes is finite, linking prime ideal finiteness to semistar operations.

Findings

Finite minimal prime sets under specific conditions

Minimal primes are radicals of $igstar$-finite ideals

Finiteness depends on properties of semistar operations

Abstract

Let $R$ be a commutative integral domain and let $\star$ be a semistar operation of finite type on $R$, and $I$ be a quasi-$\star$-ideal of $R$. We show that, if every minimal prime ideal of $I$ is the radical of a $\star$-finite ideal, then the set $\Min(I)$ of minimal prime ideals over $I$ is finite.