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On Noetherian rings with essential socle

Published online by Cambridge University Press:  09 April 2009

Jianlong Chen ,
Nanqing Ding  and
Mohamed F. Yousif
Jianlong Chen
Affiliation:
Department of Mathematics, Nanjing UniversityNanjing 210093, China and Department of Mathematices, Southeast University, Nanjing 210096, China, e-mail: jlchen@seu.edu.cn
Nanqing Ding
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China, e-mail: nqding@nju.edu.cn
Mohamed F. Yousif
Affiliation:
Department of Mathematics, Ohio State University, Lima, Ohio 45804, USA, e-mail: yousif.1@osu.edu
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Abstract

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It is shown that if R is a right Noetherian ring whose right socle is essential as a right ideal and is contained in the left socle, then R is right Artinian. This result may be viewed as a one-sided version of a result of Ginn and Moss on two-sided Noetherian rings with essential socle. This also extends the work of Nicholson and Yousif where the same result is obtained under a stronger hypothesis. We use our work to obtain partial positive answers to some open questions on right CF, right FGF and right Johns rings.

Keywords

Noetherian ringsArtinian ringsquasi-Frobenius rings
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 1 , February 2004 , pp. 39 - 50
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer, Berlin, 1974).CrossRefGoogle Scholar
[2]Camillo, V. and Yousif, M. F., ‘Continuous rings with ACC on annihilators’, Canad. Math. Bull. 34 (1991), 462–464.CrossRefGoogle Scholar
[3]Chen, J. and Ding, N., ‘On generalizations of injectivity’, in: International Symposium on Ring Theory (eds. Birkenmeier, G. F., Park, J. K. and Park, Y. S.) (Birkhäuser, Basel, 2001) pp. 85–94.CrossRefGoogle Scholar
[4]Chen, J., Ding, N., Li, Y. and Zhou, Y., ‘On (m, n)-injectivity of modules’, Comm. Algebra 29 (2001), 5589–5603.CrossRefGoogle Scholar
[5]Dung, N. V., Huynh, D. V., Smith, P. F. and Wisbauer, R., Extending modules, Pitman Research Notes in Math. 313 (Longman, Harlow, New York, 1994).Google Scholar
[6]Faith, C., ‘Finitely embedded commutative rings’, Proc. Amer. Math. Soc. 112 (1991), 657–659.CrossRefGoogle Scholar
[7]Faith, C. and Menal, P., ‘A counter example to a conjecture of Johns’, Proc. Amer. Math. Soc. 116 (1992), 21–26.CrossRefGoogle Scholar
[8]Ginn, S. M. and Moss, P. B., ‘Finitely embedded modules over Noetherian rings’, Bull. Amer. Math. Soc. 81 (1975), 709–710.CrossRefGoogle Scholar
[9]Pardo, J. L. Gómez and Asensio, P. A. Guil, ‘Torsionless modules and rings with finite essential socle’, in: Abelian groups, module theory, and topology (Padua, 1997), Lecture Notes in Pure and Appl. Math. 201 (Marcel Dekker, New York, 1998) pp. 261–278.Google Scholar
[10]Goodearl, K. R., Ring theory: nonsingular rings and modules, Monographs Textbooks Pure Appl. Math. 33 (Marcel Dekker, New York, 1976).Google Scholar
[11]Kasch, F., Modules and rings (Academic Press, London, 1982).Google Scholar
[12]Mezzetti, G., ‘Bilinear products with A.C.C. on annihilators’, Comm. Algebra 30 (2002), 1039–1047.CrossRefGoogle Scholar
[13]Nicholson, W. K. and Yousif, M. F., ‘Principally injective rings’, J. Algebra 174 (1995), 77–93.CrossRefGoogle Scholar
[14]Nicholson, W. K. and Yousif, M. F., ‘Mininjective rings’, J. Algebra 187 (1997), 548–578.CrossRefGoogle Scholar
[15]Nicholson, W. K. and Yousif, M. F., ‘On finitely embedded rings’, Comm. Algebra 28 (2000), 5311–5315.CrossRefGoogle Scholar
[16]Nicholson, W. K. and Yousif, M. F., ‘On quasi-Frobenius rings’, in: International symposium on ring theory (eds. Birkenmeier, G. F., Park, J. K. and Park, Y. S.) (Birkhäuser, Basel, 2001) pp. 245–277.CrossRefGoogle Scholar
[17]Nicholson, W. K. and Yousif, M. F., ‘Weakly continuous and C2-rings’, Comm. Algebra 29 (2001), 2429–2446.CrossRefGoogle Scholar
[18]Rutter, E. A. Jr, ‘Two characterizations of quasi-Frobenius rings’, Pacific J. Math. 30 (1969), 777–784.CrossRefGoogle Scholar
[19]Rutter, E. A. Jr, ‘Rings with the principal extension property’, Comm. Algebra 3 (1975), 203–212.CrossRefGoogle Scholar
[20]Wisbauer, R., Yousif, M. F. and Zhou, Y., ‘Ikeda-Nakayama modules’, Beitraege Zur Algebra and Geometrie 43 (2002), 111–119.Google Scholar
[21]Yousif, M. F., ‘On continuous rings’, J. Algebra 191 (1997), 495–509.CrossRefGoogle Scholar