Semistar dimension of polynomial rings and Pr\"{u}fer-like domains

TL;DR

This paper introduces the semistar dimension formula for integral domains, exploring its connections with various domain classes and extending Arnold's formula within the semistar operation framework.

Contribution

It defines the semistar dimension formula, relates it to domain properties, and extends Arnold's formula to semistar operations, providing new characterizations of specific domain types.

Findings

Established the semistar dimension inequality formula.

Connected the formula with $ar{ ext{star}}$-universally catenarian domains.

Extended Arnold's formula to semistar operations.

Abstract

Let $D$ be an integral domain and $\star$ a semistar operation stable and of finite type on it. In this paper we define the semistar dimension (inequality) formula and discover their relations with $\star$-universally catenarian domains and $\star$-stably strong S-domains. As an application we give new characterizations of $\star$-quasi-Pr\"{u}fer domains and UM$t$ domains in terms of dimension inequality formula (and the notions of universally catenarian domain, stably strong S-domain, strong S-domain, and Jaffard domains). We also extend Arnold's formula to the setting of semistar operations.