# Centralizer nearrings

Document Type : Research Article

Author

Department of Mathematics Southeastern Louisiana University

Abstract

Suppose that $(G,+)$ is a group (possibly nonabelian) and that $X$ is a submonoid of the monoid of all endomorphisms of $G$ under the operation of composition of functions, $({\rm End}~{G}, \circ)$. We define the $X$-centralizer nearring of $G$ by $X$ by saying that $M_X(G):=\{ f:G \to G \mid f(0_G)=0_G \text{ and } f \circ \alpha=\alpha \circ f \text{ for all } \alpha \in X \}$. This set of functions, $M_X(G)$, is a nearring under the usual" operations of function addition" and composition" of functions. This paper investigates how centralizer nearrings can be defined and investigates their ideals when $X$ is a group of automorphisms.

Keywords

Main Subjects

#### References

[1] G. A. Cannon, Centralizer near-rings determined by EndG, Near-Rings and Near-Fields (Fredericton, NB, 1993), Math. Appl., 336, Kluwer Acad. Publ., Dordrecht, pp. 89–111, 1995.
[2] G. A. Cannon and L. Kabza, Simplicity of the centralizer near-ring determined by EndG, Algebra Colloq. 5 (1998), 4, 383–390.
[3] G. A. Cannon, K. Neuerberg and G. L. Walls, Simplicity of Full Centralizer Nearrings and Centralizer Exponent-Preserving Groups, preprint
[4] J. R. Clay, Nearrings: Geneses and Applications, Oxford University Press, Oxford, 1992.
[5] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1985.
[6] Y. Fong and J. D. P. Meldrum, Endomorphism near-rings of a direct sum of isomorphic finite simple non-abelian groups, Near-Rings and Near-Fields, G. Betsch, ed., North-Holland, Amsterdam, pp. 73–78, 1987.
[7] C. J. Maxson and A. Oswald, On the centralizer of a semigroup of group endomorphisms, Semigroup Forum 28 (1984), no. 1-3, 29–46.
[8] C. J. Maxson and K. C. Smith, The centralizer of a set of group automorphisms, Comm. Algebra 8 (1980), no. 3, 211–230.
[9] J. D. P. Meldrum, Near-Rings and Their Links with Groups, Research Notes in Math., No. 134, Pitman Publ. Co., London, 1985.
[10] G. Pilz, Near-Rings, North-Holland/American Elsevier, Amsterdam, 1983.
[11] D. J. S. Robinson, A Course in the Theory of Groups, 2nd ed., Springer-Verlag, New York, 1996.
[12] M. R. Zinov’eva and A. S. Kondrat’ev, Finite almost simple groups with prime graphs all of whose connected components are cliques, (Russian) translated from Tr. Inst. Mat. Mekh. 21 (2015), no. 3, 132–141 Proc. Steklov Inst. Math. 295 (2016), suppl. 1, S178–S188.