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Structure of Some Noetherian Injective Modules

Published online by Cambridge University Press:  20 November 2018

B. Sarath
B. Sarath*
Affiliation:
Shiraz University, Shiraz, Iran
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The main object of this paper is to study when infective noetherian modules are artinian. This question was first raised by J. Fisher and an example of an injective noetherian module which is not artinian is given in [9]. However, it is shown in [4] that over commutative rings, and over hereditary noetherian P.I rings, injective noetherian does imply artinian. By combining results of [6] and [4] it can be shown that the above implication is true over any noetherian P.I ring. It is shown in this paper that injective noetherian modules are artinian over rings finitely generated as modules over their centers, and over semiprime rings of Krull dimension 1. It is also shown that every injective noetherian module over a P.I ring contains a simple submodule. Since any noetherian injective module is a finite direct sum of indecomposable injectives it suffices to study when such injectives are artinian. If Q is an injective indecomposable noetherian module, then Q contains a non-zero submodule Q0 such that the endomorphism rings of Q0 and all its submodules are skewfields. Over a commutative ring, such a Q0 is simple. In the non-commutative case very little can be concluded, and many of the difficulties seem to arise here.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 6 , 01 December 1980 , pp. 1277 - 1287
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Bass, H., Finitistic homological dimension and a homological generalisation of semiprimary rings, Trans. A.M.S. 95 (1960), 466–488.Google Scholar
2. Cozzens, J. H., Homological properties of the ring of differential polynomials, Bull. A.M.S. 76 (1970), 75–99.Google Scholar
3. Fisher, J., Nil subrings of endomorphism rings of modules, Proc. A.M.S. 34 (1972), 75–78.Google Scholar
4. Fisher, J., Finiteness conditions for projective and injective modules, Proc. A.M.S. 40 (1973), 389–394.Google Scholar
5. Gordon, R. and Robson, J., Krull dimension, Mem. A.M.S. (1970).Google Scholar
6. Jategaonkar, A. V., Notices A.M.S. 31-16-25 (1976).Google Scholar
7. Lenagan, T., Artinian ideals in noetherian rings, Proc. A.M.S. 51 (1975), 499–500.Google Scholar
8. Matlis, E., Injective modules over noetherian rings, Pac. J. Math. 8 (1958), 511–528.Google Scholar
9. Miller, R. and Turnidge, D., Some examples from infinite matrix rings, Proc. A.M.S. 38 (1973), 65–67.Google Scholar
10. Procesi, C., Rings with polynomial identities (Marcel Dekker Inc., New York, 1973).Google Scholar
11. Sarath, B. and Varadarajan, K., Injectivity of direct sums, Comm. in Alg. 1 (1974), 517–530.Google Scholar
12. Shock, R., Certain artinian rings are noetherian, Can. J. Math. 24 (1972), 553–556.Google Scholar