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Rings whose Indecomposable Injective Modules are Uniserial

Published online by Cambridge University Press:  20 November 2018

David A. Hill
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A module is uniserial in case its submodules are linearly ordered by inclusion. A ring R is left (right) serial if it is a direct sum of uniserial left (right) R-modules. A ring R is serial if it is both left and right serial. It is well known that for artinian rings the property of being serial is equivalent to the finitely generated modules being a direct sum of uniserial modules [8]. Results along this line have been generalized to more arbitrary rings [6], [13].

This article is concerned with investigating rings whose indecomposable injective modules are uniserial. The following question is considered which was first posed in [4]. If an artinian ring R has all indecomposable injective modules uniserial, does this imply that R is serial? The answer is yes if R is a finite dimensional algebra over a field. In this paper it is shown, provided R modulo its radical is commutative, that R has every left indecomposable injective uniserial implies that R is right serial.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 4 , 01 August 1982 , pp. 797 - 805
Copyright
Copyright © Canadian Mathematical Society 1982

References

References>

1. Anderson, F. W. and Fuller, K. R., Rings and categories of modules (Springer-Verlag, New York-Heidelberg-Berlin, 1973).Google Scholar
2. Camillo, V. P. and Fuller, K. R., Balanced and QF-\ algebras, Proc. Amer. Math. Soc. 34 (1972), 373–378.Google Scholar
3. Dlab, V. and Ringel, C. M., Decomposition of modules over right serial rings, Math. Z. 129 (1972), 207–230.Google Scholar
4. Fuller, K. R., On indecomposable infectives over artinian rings, Pacific J. Math. 29 (1969), 115–135.Google Scholar
5. Fuller, K. R., On direct representations of quasi-injectives and quasi-projectives, Arch. Math. 20 (1969), 495–503.Google Scholar
6. Ivanov, G., Decomposition of modules over serial rings, Comm. in Alg. 3 (1975), 1031–1036.Google Scholar
7. Nakayama, T., On frobeniusean algebras I, Ann. of Math. 40 (1939), 611–634.Google Scholar
8. Nakayama, T., On frobeniusean algebras II, Ann. of Math. 42 (1941), 1–21.Google Scholar
9. Nakayama, T., Note on uniserial and generalized uniserial rings, Proc. Imp. Soc. Japan 16 (1940), 285–289.Google Scholar
10. Robert, E., Projectifs et injectifs relatifs, C. R. Acad. Sci. Paris. Ser. A 268 (1969), 361–364.Google Scholar
11. Roux, B., Anneaux artiniens et extensions d'anneaux semi-simples, J. Algebra 25 (1973), 295–306.Google Scholar
12. Roux, B., Modules injectifs indécomposables sur les anneaux artiniens et dualité de Morita, Séminaire Dubreil (Algèbre) 26 (1973), no. 10.Google Scholar
13. Warfield, R. B. Jr., Serial rings, semi-hereditary rings and finitely presented modules, J. Algebra 37 (1975), 187–222.Google Scholar