On the structure of zero-divisor elements in near-ring of skew formal power series

Commun. Korean Math. Soc. Published online February 9, 2021

Abdollah Alhevaz, Ebrahim Hashemi, and Fatemeh Shokuhifar
Shahrood University of Technology

Abstract : The main purpose of this paper is to study the zero-divisor properties of the zero-symmetric near-ring of skew formal power series $ R_{0}[[x;\alpha]] $, where $ R $ is a symmetric, $ \alpha $-compatible and right Noetherian ring. It is shown that if $ R $ is reduced, then the set of all zero-divisor elements of $ R_{0}[[x;\alpha]] $ forms an ideal of $ R_{0}[[x;\alpha]] $ if and only if $ Z(R) $ is an ideal of $ R $. Also, if $ R $ is a non-reduced ring and $ ann_{R}( a-b)\cap Nil(R)\neq 0 $ for each $ a,b\in Z(R) $, then $ Z\big(R_{0}[[x;\alpha]]\big)$ is an ideal of $ R_{0}[[x;\alpha]] $. Moreover, if $ R $ is a non-reduced right Noetherian ring and $ Z\big(R_{0}[[x;\alpha]]\big)$ forms an ideal, then $ ann_{R}( a-b)\cap Nil(R)\neq 0 $ for each $ a,b\in Z(R) $. Also, it is proved that the only possible diameters of the zero-divisor graph of
$R_{0}[[x;\alpha]]$ is 2 and 3.

Keywords : Symmetric ring, $ \alpha $-compatible ring, near-ring of skew formal power series, zero-divisor element