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Rings with finite essential socle

Authors: José L. Gómez Pardo and Pedro A. Guil Asensio
Journal: Proc. Amer. Math. Soc. 125 (1997), 971-977
MSC (1991): Primary 16L30; Secondary 16D50, 16E50, 16L60, 16S50
MathSciNet review: 1371138
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Abstract: Let $R$ be a ring such that every direct summand of the injective envelope $E=E(R_R)$ has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right $R$-modules is no larger than that of the isomorphism classes of minimal right ideals, then $R_R$ cogenerates the simple right $R$-modules and has finite essential socle. This extends Osofsky's theorem which asserts that a right injective cogenerator ring has finite essential right socle. It follows from our result that if $R_R$ is a CS cogenerator, then $R_R$ is already an injective cogenerator and, more generally, that if $R_R$ is CS and cogenerates the simple right $R$-modules, then it has finite essential socle. We show with an example that in the latter case $R_R$ need not be an injective cogenerator.

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Additional Information

José L. Gómez Pardo
Affiliation: Departamento de Alxebra, Universidade de Santiago, 15771 Santiago de Compostela, Spain

Pedro A. Guil Asensio
Affiliation: Departamento de Matematicas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain

Received by editor(s): September 28, 1995
Additional Notes: This work was partially supported by the DGICYT (PB93-0515, Spain). The first author was also partially supported by the European Community (Contract CHRX-CT93-0091) and the Xunta de Galicia (XUGA 10502B94), and the second author by the C. A. de Murcia (PIB 94-25).
Communicated by: Ken Goodearl
Article copyright: © Copyright 1997 American Mathematical Society

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