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4 - Min-CS Rings

Published online by Cambridge University Press:  14 September 2009

W. K. Nicholson
Affiliation:
University of Calgary
M. F. Yousif
Affiliation:
Ohio State University
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Summary

In this chapter, we consider the class of left min-CS rings (for which every minimal left ideal is essential in a direct summand) and show that this weak injectivity property is useful in obtaining semiperfect rings. Indeed, it is proved in Theorem 4.8 that if R is left min-CS, then the dual of every simple right R-module is simple, if and only if R is semiperfect with Sι = Sr and soc(Re) is simple and essential for every local idempotent e of R. The hypotheses of Theorem 4.8 are the weakest known conditions of this type that imply that R is semiperfect.

If we strengthen the left min-CS hypothesis in Theorem 4.8 by requiring that each closed left ideal with simple essential socle be a direct summand of RR (R is left strongly min-CS), we obtain a class of rings that satisfies many of the characteristic properties of left PF rings. If instead of assuming in Theorem 4.8 that the duals of simple right R-modules are simple we suppose, more generally, that R is right Kasch, then we obtain a larger class of rings that still retains many of these properties: It is shown in Theorem 4.10 that R is left CS and right Kasch if and only if it is semiperfect and left continuous with SressRR.

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Publisher: Cambridge University Press
Print publication year: 2003

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  • Min-CS Rings
  • W. K. Nicholson, University of Calgary, M. F. Yousif, Ohio State University
  • Book: Quasi-Frobenius Rings
  • Online publication: 14 September 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511546525.005
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  • Min-CS Rings
  • W. K. Nicholson, University of Calgary, M. F. Yousif, Ohio State University
  • Book: Quasi-Frobenius Rings
  • Online publication: 14 September 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511546525.005
Available formats
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To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Min-CS Rings
  • W. K. Nicholson, University of Calgary, M. F. Yousif, Ohio State University
  • Book: Quasi-Frobenius Rings
  • Online publication: 14 September 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511546525.005
Available formats
×