[Paper Review] Uppers to zero in polynomial rings and Prüfer-like domains

Gyu Whan Chang, Marco Fontana|ArXiv.org|Jan 10, 2008

Rings, Modules, and Algebras34 references12 citations

TL;DR

This paper introduces and characterizes $φ$-quasi-Pr"ufer domains using semistar operations, generalizing classical notions like quasi-Pr"ufer and UM$t$-domains. It establishes that a domain is $φ_{f}$-quasi-Pr"ufer if and only if it is a $φ_{f}$-primitive extension, and proves that the integral closure of a UM$t$-domain is a P$v$MD if and only if the $w$-closure of $D$ is a P$v$MD and the $w$-operations on $D$ and its closure are compatible via the canonical embedding.

ABSTRACT

Let $D$ be an integral domain and $X$ an indeterminate over $D$. It is well known that (a) $D$ is quasi-Prüfer (i.e, its integral closure is a Prüfer domain) if and only if each upper to zero $Q$ in $D[X] $ contains a polynomial $g \in D[X]$ with content $\co_D(g) = D$; (b) an upper to zero $Q$ in $D[X]$ is a maximal $t$-ideal if and only if $Q$ contains a nonzero polynomial $g \in D[X]$ with $\co_D(g)^v = D$. Using these facts, the notions of UM$t$-domain (i.e., an integral domain such that each upper to zero is a maximal $t$-ideal) and quasi-Prüfer domain can be naturally extended to the semistar operation setting and studied in a unified frame. In this paper, given a semistar operation $\star$ in the sense of Okabe-Matsuda, we introduce the $\star$-quasi-Prüfer domains. We give several characterizations of these domains and we investigate their relations with the UM$t$-domains and the Prüfer $v$-multiplication domains.

Motivation & Objective

• To extend the classical notions of quasi-Pr"ufer domains and UM$t$-domains to the semistar operation setting.
• To investigate the structural and ideal-theoretic properties of $φ$-quasi-Pr"ufer domains in relation to primitive extensions and the INC property.
• To resolve open questions regarding the integral closure of UM$t$-domains and the local-global behavior of these domains.
• To provide a unified framework for studying Pr"ufer-like domains using semistar operations.
• To prove that a UM$t$-domain has a P$v$MD integral closure if and only if its $w$-closure is a P$v$MD and the $w$-operations are compatible under the canonical embedding.

Proposed method

• Introduce the concept of $φ$-quasi-Pr"ufer domains using semistar operations $φ$ on an integral domain $D$ with quotient field $K$.
• Use the notion of $φ_f$-primitive extensions and the associated $Na(D, φ_f)$ construction to characterize $φ_f$-quasi-Pr"ufer domains.
• Establish equivalences between $φ_f$-quasi-Pr"ufer domains, $φ_f$-INC-domains, and the extended prime ideal property in $Na(D, φ_f)$.
• Apply the $w$-operation and $v$-operation to analyze dimension and integrality properties, particularly in the context of UM$t$-domains.
• Use the canonical embedding $ ilde{\iota}: D \hookrightarrow \widetilde{D}$ to relate the $w$-operations on $D$ and its integral closure $\widetilde{D}$, proving $(w_D)_{\widetilde{\iota}} = w_{\widetilde{D}}$.
• Leverage known results on P$v$MDs and $t$-invertibility to show that $D$ is a P$\star$MD if and only if every two-generated ideal is $\star_f$-invertible.

Experimental results

Research questions

• RQ1 When is a domain $D$ a $\star_f$-quasi-Pr"ufer domain, and how does this relate to $\star_f$-primitive extensions and the INC property?
• RQ2 What conditions ensure that the integral closure of a UM$t$-domain is a P$v$MD, and how is this related to the behavior of $w$-operations?
• RQ3 How do the $w$-dimension and $t$-dimension of a UM$t$-domain relate to the dimension of its $Na(D,v)$ ring?
• RQ4 In what way does the semistar operation framework unify the theory of quasi-Pr"ufer domains, UM$t$-domains, and P$v$MDs?
• RQ5 Is the local-global behavior of UM$t$-domains fully characterized by the compatibility of $w$-operations on $D$ and its integral closure?

Key findings

• A domain $D$ is $\star_f$-quasi-Pr"ufer if and only if $D \subseteq K$ is a $\star_f$-primitive extension.
• A domain $D$ is $\star_f$-quasi-Pr"ufer if and only if every overring $R$ of $D$ is a $(\star_f)_{\iota}$-quasi-Pr"ufer domain.
• The ring $\mathrm{Na}(D, \star_f)$ is a quasi-Pr"ufer domain if and only if $D$ is $\star_f$-quasi-Pr"ufer.
• The integral closure of $\mathrm{Na}(D, \star_f)$ is a Pr"ufer domain if and only if $D$ is $\star_f$-quasi-Pr"ufer.
• For a semistar operation $\star$, $D$ is $\star_f$-quasi-Pr"ufer if and only if $D$ is $t$-quasi-Pr"ufer and every $\star_f$-maximal ideal is a $t$-ideal.
• A UM$t$-domain $D$ has a P$v$MD integral closure if and only if $\widetilde{D}$ is a P$v$MD and $(w_D)_{\widetilde{\iota}} = w_{\widetilde{D}}$, resolving an open problem on the local-global behavior of UM$t$-domains.