Prime Avoidance Property

TL;DR

This paper characterizes when subsets of prime spectra in commutative rings have the prime avoidance property, linking it to compactness and providing conditions and counterexamples across various classical rings.

Contribution

It precisely determines conditions for prime avoidance property in subsets of spectra and explores its implications and limitations in classical ring contexts.

Findings

Prime avoidance property is characterized for subsets of Spec(R).

If a subset has prime avoidance, it is necessarily compact.

Counterexamples show the property does not always hold in Prufer domains.

Abstract

Let $R$ be a commutative ring, we say that $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property, if $I\subseteq \bigcup_{P\in\mathcal{A}}P$ for an ideal $I$ of $R$, then there exists $P\in\mathcal{A}$ such that $I\subseteq P$. We exactly determine when $\mathcal{A}\subseteq Spec(R)$ has prime avoidance property. In particular, if $\mathcal{A}$ has prime avoidance property, then $\mathcal{A}$ is compact. For certain classical rings we show the converse holds (such as Bezout rings, $QR$-domains, zero-dimensional rings and $C(X)$). We give an example of a compact set $\mathcal{A}\subseteq Spec(R)$, where $R$ is a Prufer domain, which has not $P.A$-property. Finally, we show that if $V,V_1,\ldots, V_n$ are valuation domains for a field $K$ and $V[x]\nsubseteq \bigcup_{i=1}^n V_i$ for some $x\in K$, then there exists $v\in V$ such that $v+x\notin \bigcup_{i=1}^n V_i$.