On the structure of zero-divisor elements in a near-ring of skew formal power series
Commun. Korean Math. Soc. 2021 Vol. 36, No. 2, 197-207
https://doi.org/10.4134/CKMS.c190433
Published online February 9, 2021
Printed April 30, 2021
Abdollah Alhevaz, Ebrahim Hashemi, Fatemeh Shokuhifar
Shahrood University of Technology; Shahrood University of Technology; 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
MSC numbers : Primary 16Y30, 16U99; Secondary 05C12
Supported by : This research was in part supported by a grant from Shahrood University of Technology.
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