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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Injective cogenerator rings and a theorem of Tachikawa
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by Carl Faith PDF
Proc. Amer. Math. Soc. 60 (1976), 25-30 Request permission

Abstract:

Tachikawa showed that a left perfect ring R is an injective cogenerator in the category of all right R-modules iff there holds: (right FPF) every finitely generated faithful right module generates $\bmod {\text {-}}R$. In this paper, we simplify Tachikawa’s long and difficult proof by first obtaining some new structure theorems for a general semiperfect right FPF ring R; the most important are: R is a direct sum of uniform right ideals, and every nonzero right ideal of the basic ring ${R_0}$ of R contains a nonzero ideal of ${R_0}$. Furthermore, if the Jacobson radical rad R is nil, then R is right self-injective. Tachikawa’s theorem is an immediate consequence. We also generalize a theorem of Osofsky on perfect PF rings to FPF rings.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 60 (1976), 25-30
  • MSC: Primary 16A36
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0417237-4
  • MathSciNet review: 0417237