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Injective Modules Over Non-Artinian Serial Rings

Published online by Cambridge University Press:  09 April 2009

David A. Hill
David A. Hill
Affiliation:
Instituto de MatematicaUniversidade Federal de Bahia Salvador, Bahia Brasil
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Abstract

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A module is uniserial if its lattice of submodules is linearly ordered, and a ring R is left serial if R is a direct sum of uniserial left ideals. The following problem is considered. Suppose the injective hull of each simple left R-module is uniserial. When does this imply that the indecomposable injective left R-modules are uniserial? An affirmative answer is known when R is commutative and when R is Artinian. The following result is proved.

Let R be a left serial ring and suppose that for each primitive idempotent e, eRe has indecomposable injective left modules uniserial. The following conditions are equivalent. (a) The injective hull of each simple left R-module is uniserial. (b) Every indecomposable injective left R-module is univerial. (c) Every finitely generated left R-module is serial.

The rest of the paper is devoted to a study of some non-Artinian serial rings which serve to illustrate this theorem.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 44 , Issue 2 , April 1988 , pp. 242 - 251
Copyright
Copyright © Australian Mathematical Society 1988

References

[1]Faith, C., ‘Self injective rings’, Proc. Amer. Math. Soc. 77 (2) (1979), 157–164.Google Scholar
[2]Fuller, K. R., ‘On indecomposable injectives over Artinian rings’, Pacific J. Math. 29 (1969), 115–135.Google Scholar
[3]Gill, D. T., ‘Almost maximal valuation rings’, J. London Math. Soc. (1) 4 (1971), 140–146.Google Scholar
[4]Ivanov, G., ‘Decomposition of modules over serial rings’, Comm. Algebra 3 (1975), 1031–1036.Google Scholar
[5]Ivanov, G., ‘Left serial rings which are left Noetherian’, unpublished.Google Scholar
[6]Roux, B., ‘Sur la dualité de Morita’, Tôhoku Math. J 23 (1971), 457–472.CrossRefGoogle Scholar
[7]Singh, S., ‘Quasi-injective and quasi-projective modules over hereditary Noetherian prime rings’, Canad. J. Math. 26 (1974), 1173–1185.Google Scholar
[8]Warfield, R. B. Jr, ‘Serial rings and finitely presented modules’, J. Algebra 37 (1975), 187–222.Google Scholar