Noncommutative generalizations of theorems of Cohen and Kaplansky

TL;DR

This paper extends classical theorems of Cohen and Kaplansky to noncommutative rings, showing properties like finite generation and principality can be tested on specific sets of prime right ideals, with applications to noetherian rings.

Contribution

It introduces new noncommutative generalizations of Cohen's and Kaplansky's theorems, linking properties of rings to their prime right ideals and providing novel characterizations.

Findings

Right ideal properties can be tested on prime right ideals.

A noetherian ring is a principal right ideal ring iff all maximal right ideals are principal.

Counterexample shows left noetherian hypothesis is necessary.

Abstract

This paper investigates situations where a property of a ring can be tested on a set of "prime right ideals." Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every "prime right ideal" is finitely generated (resp. principal), where the phrase "prime right ideal" can be interpreted in one of many different ways. We also use our methods to show that other properties can be tested on special sets of right ideals, such as the right artinian property and various homological properties. Applying these methods, we prove the following noncommutative generalization of a result of Kaplansky: a (left and right) noetherian ring is a principal right ideal ring iff all of its maximal right ideals are principal. A counterexample shows that the left noetherian hypothesis cannot be dropped. Finally, we compare our results to earlier generalizations of Cohen's and Kaplansky's theorems in the literature.