## Rings with finite essential socle

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- by José L. Gómez Pardo and Pedro A. Guil Asensio
- Proc. Amer. Math. Soc.
**125**(1997), 971-977 - DOI: https://doi.org/10.1090/S0002-9939-97-03747-7
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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.## References

- Jan-Erik Björk,
*Radical properties of perfect modules*, J. Reine Angew. Math.**253**(1972), 78–86. MR**313309**, DOI 10.1515/crll.1972.253.78 - A. W. Chatters and C. R. Hajarnavis,
*Rings in which every complement right ideal is a direct summand*, Quart. J. Math. Oxford Ser. (2)**28**(1977), no. 109, 61–80. MR**437595**, DOI 10.1093/qmath/28.1.61 - Nguyen Viet Dung, Dinh Van Huynh, Patrick F. Smith, and Robert Wisbauer,
*Extending modules*, Pitman Research Notes in Mathematics Series, vol. 313, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1994. With the collaboration of John Clark and N. Vanaja. MR**1312366** - Carl Faith,
*Algebra. II*, Grundlehren der Mathematischen Wissenschaften, No. 191, Springer-Verlag, Berlin-New York, 1976. Ring theory. MR**0427349**, DOI 10.1007/978-3-642-65321-6 - Patrick J. Fleury (ed.),
*Advances in noncommutative ring theory*, Lecture Notes in Mathematics, vol. 951, Springer-Verlag, Berlin-New York, 1982. MR**672800** - J. L. Gómez Pardo and P. A. Guil Asensio,
*Essential embedding of cyclic modules in projectives*, Trans. Amer. Math. Soc., to appear. - K. R. Goodearl,
*von Neumann regular rings*, Monographs and Studies in Mathematics, vol. 4, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR**533669** - Pere Menal,
*On the endomorphism ring of a free module*, Publ. Sec. Mat. Univ. Autònoma Barcelona**27**(1983), no. 1, 141–154. MR**763863** - B. L. Osofsky,
*A generalization of quasi-Frobenius rings*, J. Algebra**4**(1966), 373–387. MR**204463**, DOI 10.1016/0021-8693(66)90028-7 - J. Rada and M. Saorín,
*On semiregular rings whose finitely generated modules embed in free modules*, Canad. Math. Bull., to appear. - S. Tariq Rizvi,
*Commutative rings for which every continuous module is quasi-injective*, Arch. Math. (Basel)**50**(1988), no. 5, 435–442. MR**942540**, DOI 10.1007/BF01196504 - B. Stenström,
*Rings of Quotients*, Springer-Verlag, Berlin and New York, 1975. - R. B. Warfield Jr.,
*Serial rings and finitely presented modules*, J. Algebra**37**(1975), no. 2, 187–222. MR**401836**, DOI 10.1016/0021-8693(75)90074-5

## Bibliographic 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
- 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
- © Copyright 1997 American Mathematical Society
- 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