Avoidance and Absorbance

TL;DR

This paper explores the dual concepts of prime avoidance and absorbance, generalizing the prime avoidance lemma to radical ideals and establishing criteria for rings to satisfy these properties, especially in relation to chain conditions.

Contribution

It introduces new criteria for rings to be C.P. and P.Z., extending prime avoidance and absorbance concepts to radical ideals and analyzing their interaction with chain conditions.

Findings

A ring is C.P. and P.Z. iff it has finite spectrum.

Generalization of prime avoidance lemma to radical ideals.

Criteria for rings to satisfy avoidance and absorbance properties.

Abstract

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be C.P. (every set of primes satisfies avoidance) or P.Z. (every set of primes satisfies absorbance). Special consideration is given to the interaction with chain conditions and Noetherian-like properties. It is shown that a ring is both C.P. and P.Z. iff it has finite spectrum.