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Centralizer near-rings that are endomorphism rings


Authors: Carlton J. Maxson and Kirby C. Smith
Journal: Proc. Amer. Math. Soc. 80 (1980), 189-195
MSC: Primary 16A76; Secondary 16A44
DOI: https://doi.org/10.1090/S0002-9939-1980-0577742-8
MathSciNet review: 577742
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Abstract: For a finite ring R with identity and a finite unital R-module V the set $ C(R;V) = \{ f:V \to V\vert f(\alpha v) = \alpha f(v)$ for all $ \alpha \in R,v \in V\} $ is the centralizer near-ring determined by R and V. Those rings R such that $ C(R;V)$ is a ring for every R-module V are characterized. Conditions are given under which $ C(R;V)$ is a semisimple ring. It is shown that if $ C(R;V)$ is a semisimple ring then $ C(R;V) = {\text{End}_R}(V)$.


References [Enhancements On Off] (What's this?)

  • [1] G. Betsch, Some structure theorems on 2-primitive near-rings, Rings, Modules and Radicals (Proc. Colloq., Keszthely, 1971), Colloq. Math. Soc. János Bolyai, Vol. 6, North-Holland, Amsterdam, 1973, pp. 73-102. MR 0360722 (50:13169)
  • [2] C. J. Maxson and K. C. Smith, The centralizer of a group automorphism, J. Algebra 54 (1978), 27-41. MR 511455 (80b:16029)
  • [3] -, The centralizer of a group endomorphism, J. Algebra 57 (1979), 441-448. MR 533806 (80j:20031)
  • [4] -, The centralizer of a set of group automorphisms, Comm. Algebra 8 (1980), 211-230. MR 558111 (81c:16048)
  • [5] -, Simple near-ring centralizers of finite rings, Proc. Amer. Math. Soc. 75 (1979), 8-12. MR 529202 (81c:16047)
  • [6] -, Near-ring centralizers, Proc. Ninth Annual USL Math. Conf. Research Series 48, Univ. Southwestern Louisiana, April, 1979, pp. 49-58.
  • [7] B. R. McDonald, Finite rings with identity, Dekker, New York, 1974. MR 0354768 (50:7245)
  • [8] G. Pilz, Near-rings, The Theory and its Applications, North-Holland Math. Studies, No. 23, North-Holland, Amsterdam, New York and Oxford, 1977. MR 0469981 (57:9761)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1980-0577742-8
Keywords: Centralizers, near-rings, semisimple rings
Article copyright: © Copyright 1980 American Mathematical Society

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