A one-sided Prime Ideal Principle for noncommutative rings

TL;DR

This paper introduces completely prime right ideals as a noncommutative analogue of prime ideals, proves a key principle relating maximal right ideals to complete primeness, and explores their applications in understanding ring structure.

Contribution

It establishes the Completely Prime Ideal Principle for noncommutative rings and investigates the properties and applications of completely prime right ideals.

Findings

Maximal right ideals in a specific sense are completely prime.

Completely prime right ideals influence the one-sided structure of rings.

The principle recovers results about noncommutative domains and ring conditions.

Abstract

Completely prime right ideals are introduced as a one-sided generalization of the concept of a prime ideal in a commutative ring. Some of their basic properties are investigated, pointing out both similarities and differences between these right ideals and their commutative counterparts. We prove the Completely Prime Ideal Principle, a theorem stating that right ideals that are maximal in a specific sense must be completely prime. We offer a number of applications of the Completely Prime Ideal Principle arising from many diverse concepts in rings and modules. These applications show how completely prime right ideals control the one-sided structure of a ring, and they recover earlier theorems stating that certain noncommutative rings are domains (namely, proper right PCI rings and rings with the right restricted minimum condition that are not right artinian). In order to provide a deeper understanding of the set of completely prime right ideals in a general ring, we study the special subset of comonoform right ideals.