On $w$-Hilbert domains

TL;DR

This paper introduces the concept of $w$-Hilbert domains, explores their properties and relationships with other domain types, and characterizes when polynomial and Anderson rings are $w$-Hilbert domains.

Contribution

It defines $w$-Hilbert domains, analyzes their properties, and provides criteria for polynomial and Anderson rings to be $w$-Hilbert domains, expanding the understanding of domain classifications.

Findings

$w$-Hilbert domains are related to Hilbert, Mori, and UMT domains.

Necessary and sufficient conditions for polynomial rings to be $w$-Hilbert.

Comparison of $w$-dimension between polynomial rings and base rings.

Abstract

In this paper, we introduce the notion of a $w$-Hilbert domain and investigate its basic properties. More precisely, we explore its relationship with Hilbert domains, strong Mori domains, and UMT domains by providing various examples using $D+M$ constructions. Furthermore, we establish necessary and sufficient conditions for the polynomial ring and the Anderson ring to be $w$-Hilbert domains, and compare the $w$-dimension of the polynomial ring with that of its base ring.