Noncommutative Generalizations of Theorems of Cohen and Kaplansky
 Manuel L. Reyes
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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.
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 Title
 Noncommutative Generalizations of Theorems of Cohen and Kaplansky
 Open Access
 Available under Open Access This content is freely available online to anyone, anywhere at any time.
 Journal

Algebras and Representation Theory
Volume 15, Issue 5 , pp 933975
 Cover Date
 20121001
 DOI
 10.1007/s1046801192737
 Print ISSN
 1386923X
 Online ISSN
 15729079
 Publisher
 Springer Netherlands
 Additional Links
 Topics
 Keywords

 Point annihilators
 Cocritical right ideals
 Cohen’s theorem
 Right noetherian rings
 Kaplansky’s theorem
 Principal right ideals
 Primary 16D25
 16P40
 16P60; Secondary 16N60
 Authors

 Manuel L. Reyes ^{(1)}
 Author Affiliations

 1. Department of Mathematics, University of California, Berkeley, CA, 947203840, USA