Indecomposable decomposition and couniserial dimension

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• 作者：A. Ghorbani (1)
  S. K. Jain (2) (3)
  Z. Nazemian (1)
  
  1. Department of Mathematical Sciences ; Isfahan University of Technology ; P.O.Box ; 84156-83111 ; Isfahan ; Iran
  2. Department of Mathematics ; Ohio University ; Athens ; OH ; 45701 ; USA
  3. King Abdulaziz University ; Jeddah ; Saudi Arabia
  
• 关键词：Uniform module ; Maximal right quotient ring ; Indecomposable decomposition ; Uniserial dimension ; Couniserial dimension ; Von Neumann regular ring ; Semisimple module ; Primary 16D70 ; 16D90 ; 16P70 ; Secondary 03E10 ; 13E10
• 刊名：Bulletin of Mathematical Sciences
• 出版年：2015
• 出版时间：April 2015
• 年：2015
• 卷：5
• 期：1
• 页码：121-136
• 全文大小：198 KB
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• 刊物主题：Mathematics, general;
• 出版者：Springer Basel
• ISSN：1664-3615

文摘

Dimensions like Gelfand, Krull, Goldie have an intrinsic role in the study of theory of rings and modules. They provide useful technical tools for studying their structure. We define and study couniserial dimension for modules. Couniserial dimension is a measure of how far a module deviates from being uniform. Despite their different objectives, it turns out that there are certain common properties between the couniserial dimension and Krull dimension. Among others, each module having such a dimension contains a uniform submodule and has finite uniform dimension. Like all dimensions, this is an ordinal valued invariant. Every module of finite length has couniserial dimension and its value lies between the uniform dimension and the length of the module. Modules with countable couniserial dimension are shown to possess indecomposable decomposition. In particular, a von Neumann regular ring with countable couniserial dimension is semisimple artinian. If the maximal right quotient ring of a semiprime right non-singular ring \(R\) has a couniserial dimension as an \(R\) -module, then \(R\) is a semiprime right Goldie ring. As one of the applications, it follows that all right \(R\) -modules have couniserial dimension if and only if \(R\) is a semisimple artinian ring.