On continuous rings. (English) Zbl 0884.16002

From the abstract: We show that if \(R\) is a semiperfect ring with essential left socle and \(rl(K)=K\) for every small right ideal \(K\) of \(R\), then \(R\) is right continuous. Accordingly some well-known classes of rings, such as dual rings all of whose cyclic right \(R\)-modules are essentially embedded in projectives, are shown to be continuous. We also prove that a ring \(R\) has a perfect duality if and only if the dual of every simple right \(R\)-module is simple and \(R\oplus R\) (a direct sum) is a left and right CS module. In Section 2 of the paper we provide a characterization for semiperfect right self-injective rings in terms of the CS-condition.


16D50 Injective modules, self-injective associative rings
16L30 Noncommutative local and semilocal rings, perfect rings
Full Text: DOI


[1] Anderson, F.W.; Fuller, K.R., Rings and categories of modules, (1974), Springer-Verlag Berlin/New York · Zbl 0242.16025
[2] Björk, J.-E., Rings satisfying certain chain conditions, J. reine angew. math., 245, 63-73, (1970) · Zbl 0211.36401
[3] Camillo, V.; Yousif, M.F., Continuous rings with acc on annihilators, Canad. math. bull., 34, 462-464, (1991) · Zbl 0767.16004
[4] Dung, N.v.; Huynh, D.v.; Smith, P.F.; Wisbauer, R., Extending modules, (1994), Longman Harlow/New York
[5] Dung, N.v.; Smith, P.F., Σ-CS modules, Comm. algebra, 22, 83-93, (1994) · Zbl 0804.16001
[6] Faith, C., Rings with ascending chain condition on annihilators, Nagoya math. J., 27, 179-191, (1966) · Zbl 0154.03001
[7] Faith, C., Algebra II ring theory, (1976), Springer-Verlag Berlin/New York · Zbl 0335.16002
[8] Faticoni, T.G., FPF rings I: the Noetherian case, Comm. algebra, 13, 2119-2136, (1985) · Zbl 0579.16008
[9] J. L. Gómez Pardo, P. A. Guil Asensio, Essential embedding of cyclic modules in projectives, Trans. Amer. Math. Soc.
[10] Gómez Pardo, J.L.; Guil Asensio, P.A., Rings with finite essential socle, Proc. amer. math. soc., 125, 971-977, (1997) · Zbl 0871.16012
[11] M. Harada, Non-Small Modules and Non-Cosmall Modules, Ring Theory: Proceedings of the 1978 Antwerp Conference, F. Van Oystaeyen, Dekker, New York
[12] M. Harada, On almost relative injective of finite length · Zbl 0753.16002
[13] Huynh, D.v., A right countably sigma-CS ring with ACC or DCC on projective principal right ideals is left Artinian and QF-3, Trans. amer. math. soc., 347, 3131-3139, (1995) · Zbl 0852.16013
[14] Jain, S.K.; López-Permouth, S.R., A generalization of the wedderburn – artin theorem, Proc. amer. math. soc., 106, 19-23, (1989) · Zbl 0681.16016
[15] Jain, S.K.; López-Permouth, S.R., Rings whose cyclics are essentially embeddable in projective modules, J. algebra, 128, 257-269, (1990) · Zbl 0698.16012
[16] Mohamed, S.H.; Müller, B.J., Continuous and discrete modules, London math. soc. lecture note ser., No. 147, (1990), Cambridge Univ. Press Cambridge
[17] Nicholson, W.K., Semiregular modules and rings, Canad. J. math., 28, 1105-1120, (1976) · Zbl 0317.16005
[18] Nicholson, W.K.; Yousif, M.F., Principally injective rings, J. algebra, 174, 77-93, (1995) · Zbl 0839.16004
[19] Nicholson, W.K.; Yousif, M.F., Mininjective rings, J. algebra, 187, 548-578, (1997) · Zbl 0879.16002
[20] Oshiro, K., Lifting modules, extending modules and their applications to QF-rings, Hokkaido math. J., 13, 310-338, (1984) · Zbl 0559.16013
[21] Osofsky, B.L., A generalization of quasi-Frobenius rings, J. algebra, 4, 373-387, (1966) · Zbl 0171.29303
[22] J. Rada, M. Saorin, On Semiregular rings whose finitely generated modules embed in free, Canad. Math. Bull. · Zbl 0916.16008
[23] Rutter, E.A., Rings with the principal extension property, Comm. algebra, 3, 203-212, (1975) · Zbl 0298.16015
[24] Utumi, Y., Self-injective rings, J. algebra, 6, 56-64, (1967) · Zbl 0161.03803
[25] Utumi, Y., On continuous rings and self-injective rings, Trans. amer. math. soc., 118, 158-173, (1965) · Zbl 0144.27301
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.