Uppers to zero and semistar operations in polynomial rings

TL;DR

This paper develops a canonical way to extend stable semistar operations from an integral domain to its polynomial ring, linking properties of the domain with those of the polynomial ring through these operations.

Contribution

It introduces a new method to define stable semistar operations of finite type on polynomial rings, connecting domain properties with polynomial ring properties and providing new interpretations of localizing systems.

Findings

D is a $ ext{ extbackslash}star$-quasi-Pr"ufer domain iff each upper to zero in D[X] is quasi-$[ ext{ extbackslash}star]$-maximal

D is Pr"ufer $ ext{ extbackslash}star$-multiplication iff D[X] is Pr"u"ffer $[ ext{ extbackslash}star]$-multiplication

New interpretation of Gabriel-Popescu localizing systems via multiplicatively closed sets in D[X]

Abstract

Given a stable semistar operation of finite type $\star$ on an integral domain $D$, we show that it is possible to define in a canonical way a stable semistar operation of finite type $[\star]$ on the polynomial ring $D[X]$, such that $D$ is a $\star$-quasi-Pr\"ufer domain if and only if each upper to zero in $D[X]$ is a quasi-$[\star]$-maximal ideal. This result completes the investigation initiated by Houston-Malik-Mott \cite[Section 2]{hmm} in the star operation setting. Moreover, we show that $D$ is a Pr\"ufer $\star$-multiplication (resp., a $\star$-Noetherian; a $\star$-Dedekind) domain if and only if $D[X]$ is a Pr\"ufer $[\star]$-multiplication (resp., a $[\star]$-Noetherian; a $[\star]$-Dedekind) domain. As an application of the techniques introduced here, we obtain a new interpretation of the Gabriel-Popescu localizing systems of finite type on an integral domain $D$ (Problem 45 of \cite{cg}), in terms of multiplicatively closed sets of the polynomial ring $D[X]$.