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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.
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참고문헌 (22)
참고문헌 신청
Abdollah Alhevaz / 2016 / An Alternative Perspective on Skew Generalized Power Series Rings / Mediterranean Journal of Mathematics 13(6):4723~4744
A. ALHEVAZ / 2013 / McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS / Journal of Algebra and Its Applications 13(02):1350083
David F. Anderson / 1999 / The Zero-Divisor Graph of a Commutative Ring / Journal of Algebra 217(2):434~447
David F. Anderson / 2007 / On the diameter and girth of a zero-divisor graph / Journal of Pure and Applied Algebra 210(2):543~550
D.D. Anderson / 1993 / Beck′s Coloring of a Commutative Ring / Journal of Algebra 159(2):500~514
A. ALHEVAZ / 2013 / McCOY PROPERTY OF SKEW LAURENT POLYNOMIALS AND POWER SERIES RINGS / Journal of Algebra and Its Applications 13(02):1350083
David F. Anderson / 1999 / The Zero-Divisor Graph of a Commutative Ring / Journal of Algebra 217(2):434~447
David F. Anderson / 2007 / On the diameter and girth of a zero-divisor graph / Journal of Pure and Applied Algebra 210(2):543~550
D.D. Anderson / 1993 / Beck′s Coloring of a Commutative Ring / Journal of Algebra 159(2):500~514
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