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On Minimax and Related Modules

Published online by Cambridge University Press:  20 November 2018

Peter Rudlof
Peter Rudlof*
Affiliation:
Mathematisches Institut der Universität, Theresienstr. 39, D 8000 München 2, Federal Republic of Germany
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Abstract

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A module M is called a minimax module, if it has a finitely generated submodule U such that M/U is Artinian. This paper investigates minimax modules and some generalized classes over commutative Noetherian rings. One of our main results is: M is minimax iff every decomposition of a homomorphic image of M is finite.

From this we deduce that:

- All couniform modules are minimax.

- All modules of finite codimension are minimax.

- Essential covers of minimax modules are minimax. With the aid of these corollaries we completely determine the structure of couniform modules and modules of finite codimension.

We then examine the following variants of the minimax property:

- replace U “ finitely generated” by U “ coatomic” (i.e. every proper submodule of U is contained in a maximal submodule);

- replace M/U “ Artinian” by M/U “ semi-Artinian” (i.e. every proper submodule of M/U contains a minimal submodule).

Keywords

13E0513C05
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 44 , Issue 1 , 01 December 1991 , pp. 154 - 166
Copyright
Copyright © Canadian Mathematical Society 1989

References

1. Bass, H., Descending chains and the Krull ordinal of commutative Noetherian rings, J. Pure Appl. Algebra 1(1971), 347–360.Google Scholar
2. Bourbaki, N., Commutative algebra. Hermann, Paris, 1972.Google Scholar
3. Goodearl, K.R. and Zimmermann-Huisgen, B., Boundedness of direct products of torsion modules, J. Pure Appl. Algebra 39(1986), 251–273.Google Scholar
4. Grzeszczuk, P.and Puczylowski, E.R., On Goldie and dual Goldie dimensions, J. Pure Appl. Algebra 31(1984), 47–54.Google Scholar
5. Matlis, E., Infective modules over Noetherian rings, Pacific J. Math. 8(1958), 511–528.Google Scholar
6. Matlis, E., 1 -dimensional Cohen-Macaulay rings. Lecture notes in Mathematics 327, Springer-Verlag, Berlin, Heidelberg, New York, 1973.Google Scholar
7. Miyashita, Y., Quasi-projective modules, perfect modules and a theorem for modular lattices, J. Fac. Sci. Hokkaido Univ. 19(1966), 86–110.Google Scholar
8. Rudlof, P., Komplementierte Moduln iiber noetherschen Ringen. Dissertation, Universität München, 1989.Google Scholar
9. Rudlof, P., On the structure ofcouniform and complemented modules, J. Pure Appl. Algebra 74(1991), 281— 305.Google Scholar
10. Takeuchi, T., On cofinite-dimensionalmodules, Hokkaido Math. J. 5(1976), 1–43.Google Scholar
11. Zariski, O.and Samuel, P., Commutative algebra, Vol. II. D. van Nostrand Company Inc., New York, London, Toronto, 1960.Google Scholar
12. Zöschinger, H., Koatomare Moduln, Math. Z. 170(1980), 221–232.Google Scholar
13. Zöschinger, H., Linear-kompakte Moduln iiber noetherschen Ringen, Arch. Math. 41(1983), 121–130.Google Scholar
14. Zöschinger, H., Minimax Moduln, J. Algebra 102(1986), 1–32.Google Scholar
15. Zöschinger, H., liber die Maximalbedingung fiir radikalvolle Untermoduln, Hokkaido Math. J. 17(1988), 101— 116.Google Scholar