TL;DR
This paper explores conditions under which infinite sets of prime ideals in a commutative ring satisfy prime avoidance, providing criteria, examples, counterexamples, and applications in constructing ring counterexamples.
Contribution
It introduces new necessary and sufficient conditions for prime avoidance of infinite prime sets and demonstrates their use in constructing counterexamples in algebraic geometry.
Findings
Identifies classes of infinite prime sets satisfying prime avoidance
Provides examples and counterexamples illustrating prime avoidance phenomena
Uses prime avoidance to construct counterexamples in rings of finite type
Abstract
We investigate prime avoidance for an arbitrary set of prime ideals in a commutative ring. Various necessary and/or sufficient conditions for prime avoidance are given, which yield natural classes of infinite sets of primes that satisfy prime avoidance. Examples and counterexamples are given throughout to illustrate the phenomena that can occur. As an application, we show how to use prime avoidance to construct counterexamples among rings essentially of finite type.
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Taxonomy
TopicsRings, Modules, and Algebras · Advanced Topics in Algebra · Commutative Algebra and Its Applications