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 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.c190433Published online February 9, 2021Printed 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. Downloads: Full-text PDF   Full-text HTML

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