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The endomorphism ring of a Δ-module over a right noetherian ring

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Abstract

LetR be a right noetherian ring. A moduleM R is called a Δ-module providedR satisfies the descending chain condition for annihilators of subsets ofM. For a Δ-module, a series 0⊂M 1⊂M 2⊂...⊂M n =M can be constructed in which the factorsM i /M i−1 are sums of, α i -semicritical modules where α1≦α2≦...≦α n . In this paper we utilize this series in studying Λ=End(M R ). It is shown that ifN={f∈Λ|Kerf is essential inM}, thenN is nilpotent. Specific bounds on the index of nilpotency are given in terms of this series. Further ifM is injective and α-smooth, the annihilators of the factors of this series are used to provide necessary and sufficient conditions for EndM R to be semisimple.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Wisconsin-Milwaukee, P.O. Box 413, 53201, Milwaukee, WI, USA

    A. K. Boyle & E. H. Feller

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  1. A. K. Boyle
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  2. E. H. Feller
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Boyle, A.K., Feller, E.H. The endomorphism ring of a Δ-module over a right noetherian ring. Israel J. Math. 45, 313–328 (1983). https://doi.org/10.1007/BF02804015

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  • DOI: https://doi.org/10.1007/BF02804015

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