Article

Algebras and Representation Theory

, Volume 15, Issue 5, pp 933-975

Open Access This content is freely available online to anyone, anywhere at any time.

Noncommutative Generalizations of Theorems of Cohen and Kaplansky

  • Manuel L. ReyesAffiliated withDepartment of Mathematics, University of California Email author 

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.

Keywords

Point annihilators Cocritical right ideals Cohen’s theorem Right noetherian rings Kaplansky’s theorem Principal right ideals

Mathematics Subject Classifications (2010)

Primary 16D25 16P40 16P60; Secondary 16N60