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On Morita duality

Published online by Cambridge University Press:  17 April 2009

Weimin Xue
Weimin Xue
Affiliation:
Department of Mathematics, Fujian Normal University, Fuzhou, Fijian 350007, People's Republic of China
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Abstract

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Modules whose nonzero endomorphisms are epimorphisms and modules whose nonzero endomorphisms are monomorphisms are considered in this paper. We prove that these two classes of modules are dual to each other via Morita duality. We also prove that a left artinian ring R with Jacobson radical J has a Morita duality if either (1) J/J2 is a central bimodule; or (2) R is artinian right duo and R/J is commutative.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 49 , Issue 1 , February 1994 , pp. 35 - 46
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Anderson, F. W. and Fuller, K. R., Rings and categories of modules, 2nd edition (Springer Verlag, Berlin, Heidelberg and New York, 1992).CrossRefGoogle Scholar
[2]Azumaya, G., ‘A duality theory for injective modules’, Amer. J. Math. 81 (1959), 249–278.CrossRefGoogle Scholar
[3]Azumaya, G., ‘Exact and serial rings’, J. Algebra 85 (1983), 477–489.CrossRefGoogle Scholar
[4]Azumaya, G., ‘Characterizations of semi-perfect and perfect modules’, Math. Z. 140 (1974), 95–103.CrossRefGoogle Scholar
[5]Cohn, P.M., ‘On a class of binomial extensions’, Illinois J. Math. 10 (1966), 418–424.CrossRefGoogle Scholar
[6]Courter, R.C., ‘Finite dimensional right duo algebras are duo’, Proc. Amer. Math. Soc. 84 (1982), 157–161.Google Scholar
[7]Fisher, J.W., ‘Nil subrings of endomorphism rings of modules’, Proc. Amer. Math. Soc. 34 (1972), 75–78.CrossRefGoogle Scholar
[8]Fisher, J.W., ‘Finiteness conditions for projective and injective modules’, Proc. Amer. Math. Soc. 40 (1973), 389–394.CrossRefGoogle Scholar
[9]Hill, D.A., ‘Endomorphism rings of hereditary modules’, Arch. Math. 28 (1977), 45–50.CrossRefGoogle Scholar
[10]Hill, D.A., ‘Rings whose indecomposable injective modules are uniserial’, Canad. J. Math. 34 (1982), 797–805.CrossRefGoogle Scholar
[11]Kasch, F., Modules and rings (Academic Press, London and New York, 1982).Google Scholar
[12]Lemonnier, B., ‘Dimension de Krull et dualite de Morita dans les extensions triangulaires’, Comm. Algebra 12 (1984), 3071–3110.CrossRefGoogle Scholar
[13]Morita, K., ‘Duality for modules and its applications to the theory of rings with minimum condition’, Tokyo Kyoiku Daigaku Ser A6 (1958), 83–142.Google Scholar
[14]Mueller, B.J., ‘On Morita duality’, Canad. J. Math. 21 (1969), 1338–1347.CrossRefGoogle Scholar
[15]Mueller, B.J., ‘Linear compactness and Morita duality’, J. Algebra 16 (1970), 60–66.CrossRefGoogle Scholar
[16]Rotman, J.J., An introduction to homological algebra (Academic Press, London and New York, 1979).Google Scholar
[17]Rosenberg, A. and Zelinsky, D., ‘Finiteness of the injective hull’, Math. Z. 70 (1959), 372–380.CrossRefGoogle Scholar
[18]Shrikhande, M.S., ‘On hereditary and cohereditary modules’, Canad. J. Math. 25 (1973), 892–896.CrossRefGoogle Scholar
[19]Tiwary, A.K. and Pandeya, B.M., ‘Modules whose nonzero endomorphisms are monomorphisms’, in Algebra and its applications, (Manocha, H.L. and Srivastava, J.B., Editors), Lecture Notes Pure Appl. Math. 91 (Marcel Dekker, New York, 1984), pp. 199–203.Google Scholar
[20]Vamos, P., ‘Rings with duality’, Proc. London Math. Soc. 35 (1977), 275–289.CrossRefGoogle Scholar
[21]Whelan, E.A., ‘Finite subnormalizing extensions of rings’, J. Algebra 101 (1986), 418–432.CrossRefGoogle Scholar
[22]Xue, Weimin, ‘Morita duality and artinian left duo rings’, Bull. Austral. Math. Soc. 39 (1989), 339–342.CrossRefGoogle Scholar
[23]Xue, Weimin, ‘A note on exact hereditary rings’, Saitama Math. J. 7 (1989), 1–4.Google Scholar
[24]Xue, Weimin, ‘Morita duality and intermediate triangular extensions’, Yokohama Math. J. 37 (1989), 57–59.Google Scholar
[25]Xue, Weimin, Rings with morita duality, Lect. Notes. Math. 1523 (Springer-Verlag, Berlin, Heidelberg, New York, 1992).CrossRefGoogle Scholar