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Kyungpook Mathematical Journal 2022; 62(3): 437-453

Published online September 30, 2022

Copyright © Kyungpook Mathematical Journal.

An Alternative Perspective of Near-rings of Polynomials and Power series

Fatemeh Shokuhifar, Ebrahim Hashemi∗ and Abdollah Alhevaz

Faculty of Mathematical Sciences, Shahrood University of Technology, Shahrood 316-3619995161, Iran
e-mail : shokuhi.135@gmail.com, eb_hashemi@yahoo.com or eb_hashemi@shahroodut.ac.ir and a.alhevaz@gmail.com or a.alhevaz@shahroodut.ac.ir

Received: June 26, 2021; Revised: January 2, 2022; Accepted: January 13, 2022

Abstract

Unlike for polynomial rings, the notion of multiplication for the near-ring of polynomials is the substitution operation. This leads to somewhat surprising results. Let S be an abelian left near-ring with identity. The relation ∼ on S defined by letting a~b if and only if annS(a)=annS(b), is an equivalence relation. The compressed zero-divisor graph ΓE(S) of S is the undirected graph whose vertices are the equivalence classes induced by ∼ on S other than [0]S and [1]S, in which two distinct vertices [a]S and [b]S are adjacent if and only if ab=0 or ba=0. In this paper, we are interested in studying the compressed zero-divisor graphs of the zero-symmetric near-ring of polynomials R0[x] and the near-ring of the power series R0[[x]] over a commutative ring R. Also, we give a complete characterization of the diameter of these two graphs. It is natural to try to find the relationship between diam(ΓE(R0[x])) and diam(ΓE(R0[[x]])). As a corollary, it is shown that for a reduced ring R, diam(ΓE(R))diam(ΓE(R0[x]))diam(ΓE(R0[[x]])).

Keywords: Near-ring of polynomials, Zero-divisor graph, Compressed zero-divisor graph, Diameter of graph, Near-ring of formal power series