Generalizations of Perfect, Semiperfect, and Semiregular Rings

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• 作者：Yiqiang Zhou
• 关键词：small submodules ; projective ; covers ; ; perfect rings ; ; semiperfect rings ; ; semiregular rings
• 刊名：Algebra Colloquium
• 出版年：2000
• 出版时间：August 2000
• 年：2000
• 卷：7
• 期：3
• 页码：305-318
• 全文大小：291 KB
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Algebra
  Algebraic Geometry
  
• 出版者：Springer-Verlag HongKong Ltd.
• ISSN：0219-1733

文摘

For a ring R and a right R-module M, a submodule N of M is said to be -small in M if, whenever N + X = M with M/X singular, we have X = M. If there exists an epimorphism p: P M such that P is projective and Ker(p) is -small in P, then we say that P is a projective -cover of M. A ring R is called -perfect (resp., -semiperfect, -semiregular) if every R-module (resp., simple R-module, cyclically presented R-module) has a projective -cover. The class of all -perfect (resp., -semiperfect, -semiregular) rings contains properly the class of all right perfect (resp., semiperfect, semiregular) rings. This paper is devoted to various properties and characterizations of -perfect, -semiperfect, and -semiregular rings. We define (R) by (R)/Soc(R_R) = Jac(R/Soc(R_R)) and show, among others, the following results:(1) (R) is the largest -small right ideal of R.(2) R is -semiregular if and only if R/(R) is a von Neumann regular ring and idempotents of R(R) lift to idempotents of R.(3) R is -semiperfect if and only if R/(R) is a semisimple ring and idempotents of R/(R) lift to idempotents of R.(4) R is -perfect if and only if R/Soc(R_R) is a right perfect ring and idempotents of R/(R) lift to idempotents of R.