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

Full-text PDF Free Access

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.

**1.**Jan-Erik Björk,*Radical properties of perfect modules*, J. Reine Angew. Math.**253**(1972), 78–86. MR**0313309****2.**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**0437595****3.**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****4.**Carl Faith,*Algebra. II*, Springer-Verlag, Berlin-New York, 1976. Ring theory; Grundlehren der Mathematischen Wissenschaften, No. 191. MR**0427349****5.**Patrick J. Fleury (ed.),*Advances in noncommutative ring theory*, Lecture Notes in Mathematics, vol. 951, Springer-Verlag, Berlin-New York, 1982. MR**672800****6.**J. L. Gómez Pardo and P. A. Guil Asensio,*Essential embedding of cyclic modules in projectives*, Trans. Amer. Math. Soc., to appear.**7.**K. R. Goodearl,*von Neumann regular rings*, Monographs and Studies in Mathematics, vol. 4, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR**533669****8.**Pere Menal,*On the endomorphism ring of a free module*, Publ. Sec. Mat. Univ. Autònoma Barcelona**27**(1983), no. 1, 141–154. MR**763863****9.**B. L. Osofsky,*A generalization of quasi-Frobenius rings*, J. Algebra**4**(1966), 373–387. MR**0204463****10.**J. Rada and M. Saorín,*On semiregular rings whose finitely generated modules embed in free modules*, Canad. Math. Bull., to appear.**11.**S. Tariq Rizvi,*Commutative rings for which every continuous module is quasi-injective*, Arch. Math. (Basel)**50**(1988), no. 5, 435–442. MR**942540**, 10.1007/BF01196504**12.**B. Stenström,*Rings of Quotients*, Springer-Verlag, Berlin and New York, 1975.**13.**R. B. Warfield Jr.,*Serial rings and finitely presented modules*, J. Algebra**37**(1975), no. 2, 187–222. MR**0401836**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
16L30,
16D50,
16E50,
16L60,
16S50

Retrieve articles in all journals with MSC (1991): 16L30, 16D50, 16E50, 16L60, 16S50

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