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

DOI:
https://doi.org/10.1090/S0002-9939-97-03747-7

MathSciNet review:
1371138

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be a ring such that every direct summand of the injective envelope has an essential finitely generated projective submodule. We show that, if the cardinal of the set of isomorphism classes of simple right -modules is no larger than that of the isomorphism classes of minimal right ideals, then cogenerates the simple right -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 is a CS cogenerator, then is already an injective cogenerator and, more generally, that if is CS and cogenerates the simple right -modules, then it has finite essential socle. We show with an example that in the latter case 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

Email:
pardo@zmat.usc.es

**Pedro A. Guil Asensio**

Affiliation:
Departamento de Matematicas, Universidad de Murcia, 30100 Espinardo, Murcia, Spain

Email:
paguil@fcu.um.es

DOI:
https://doi.org/10.1090/S0002-9939-97-03747-7

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