On the structure of zero-divisor elements in a near-ring of skew formal power series
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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.
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.
참고문헌 (22)
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