Noncommutative Generalizations of Theorems of Cohen and Kaplansky

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• 作者：Manuel L. Reyes (1)
  
• 关键词：Point annihilators ; Cocritical right ideals ; Cohen鈥檚 theorem ; Right noetherian rings ; Kaplansky鈥檚 theorem ; Principal right ideals ; Primary 16D25 ; 16P40 ; 16P60 ; Secondary 16N60
• 刊名：Algebras and Representation Theory
• 出版年：2012
• 出版时间：October 2012
• 年：2012
• 卷：15
• 期：5
• 页码：933-975
• 全文大小：641KB
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• 作者单位：Manuel L. Reyes (1)
  
  1. Department of Mathematics, University of California, Berkeley, CA, 94720-3840, USA
  
• ISSN：1572-9079

文摘

This paper investigates situations where a property of a ring can be tested on a set of 鈥減rime right ideals.鈥?Generalizing theorems of Cohen and Kaplansky, we show that every right ideal of a ring is finitely generated (resp. principal) iff every 鈥減rime right ideal鈥?is finitely generated (resp. principal), where the phrase 鈥減rime right ideal鈥?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鈥檚 and Kaplansky鈥檚 theorems in the literature.