A prime ideal principle for two-sided ideals

TL;DR

This paper introduces a general 'Prime Ideal Principle' applicable to two-sided ideals in noncommutative rings, unifying and extending classical results that maximal ideals with certain properties are prime.

Contribution

It generalizes the Prime Ideal Principle from commutative to noncommutative rings, providing a uniform method to prove that maximal ideals with specific properties are prime.

Findings

Unified proof technique for prime ideals in various ring classes

Extension of classical 'maximal implies prime' results to noncommutative settings

Application to rings with properties like polynomial identity, Artin-Rees, and noetherian conditions

Abstract

Many classical ring-theoretic results state that an ideal that is maximal with respect to satisfying a special property must be prime. We present a "Prime Ideal Principle" that gives a uniform method of proving such facts, generalizing the Prime Ideal Principle for commutative rings due to T.Y. Lam and the author. Old and new "maximal implies prime" results are presented, with results touching on annihilator ideals, polynomial identity rings, the Artin-Rees property, Dedekind-finite rings, principal ideals generated by normal elements, strongly noetherian algebras, and just infinite algebras.