Injective cogenerator rings and a theorem of Tachikawa

Faith, Carl

doi:10.1090/S0002-9939-1976-0417237-4

Injective cogenerator rings and a theorem of Tachikawa
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by Carl Faith

Proc. Amer. Math. Soc. 60 (1976), 25-30

DOI: https://doi.org/10.1090/S0002-9939-1976-0417237-4

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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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