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Distributive elements in centralizer near-rings

  • C. J. Maxson (a1) and J. D. P. Meldrum (a2)

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Let <G,+> be a group with identity 0 and let S be a semigroup of endomorphisms of G. The set Ms(G)={f:GG; f(0)=0, fσ=σf, for all σ∈S} with the operations of unction addition and composition is a zero-symmetric near-ring with identity called the centralizer near-ring determined by the pair (S, G). Centralizer near-rings have been studied for many classes of semigroups of endomorphisms. (See [8] and the references given there.) In this paper we continue these investigations into the structure of centralizer near-rings via our study of the relationship between distributive elements in Ms(G) and endomorphisms in Ms(G). More specifically, let N = Ms(G) and let Nd={fN; f(g1+g2)=fg1+fg2}, the set of distributive elements in N. Under the operation of function composition, Nd is a semigroup containing the identity map, id. Moreover, Nd contains as a submonoid = {α ∈ End G; ασ=σα for all σ∈S}. Here we determine for certain semigroups S, whether or not = Nd.

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References

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1.Betsch, G., Some structure theorems on 2-primitive near-rings, Colloq. Math. Soc. Janus Bolyai 6, Rings, Modules and Radicals (Keszthely (Hungary), 1971, North-Holland, 1973).
2.Maxson, C. J. and Smith, K. C., Simple near-ring centralizers of finite rings, Proc. Amer. Math. Soc. 75 (1979), 812.
3.Maxson, C. J., and Smith, K. C., The centralizer of a set of group automorphisms, Comm. Algebra 8 (1980), 211230.
4.Maxson, C. J., and Smith, K. C., Centralizer near-rings that are endomorphism rings, Proc. Amer. Math. Soc. 80 (1980), 189195.
5.Mcdonald, B. R., Finite Rings with Identity (Dekker, N.Y., 1974).
6.Meldrum, J. D. P., On nilpotent wreath products, Proc. Cambridge Philos. Soc. 68 (1970), 115.
7.Meldrum, J. D. P., Centralisers in wreath products, Proc. Edinburgh Math. Soc. 22 (1979), 161168.
8.Pilz, G., Near-rings (Revised Edition, North-Holland, Amsterdam, N.Y. and Oxford, 1983).
9.Robinson, D. J. S., A Course in the Theory of Groups (Grad. Texts in Math. 80, Springer, Berlin, 1982).
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Distributive elements in centralizer near-rings

  • C. J. Maxson (a1) and J. D. P. Meldrum (a2)

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