Abstract
Let R be a commutative ring with nonzero identity and Z(R) be its set of zero-divisors. The new extension of the zero-divisor graph \(\widetilde{\Gamma }(R)\) with vertices \(Z(R)^{\star }\), and two distinct vertices x and y are adjacent if and only if \(xy=0\) or \(x+y\in Z(R)\). In this article, we study, in the general case, the graph \(\widetilde{\Gamma }(R)\). For any commutative ring R, we provide sufficient and necessary conditions for \(\widetilde{\Gamma }(R)\) and \(\widetilde{\Gamma }(R[x_1,\dots , x_n])\) to be complete. At last, we present some other properties of this new extension of the zero-divisor graph.
References
Atiyah, M.F., Macdonald, I.G.: Introduction to Commutative Algebra. Addison-Wesley Publishing Company, Boston (1969)
Anderson, D.F., Badawi, A.: The total graph of a commutative ring. J. Algebra 320, 2706–2719 (2008)
Anderson, D.F., Livingston, P.S.: The zero-divisor graph of a commutative ring. J. Algebra 217, 434–447 (1999)
Beck, I.: Coloring of commutative rings. J. Algebra 116, 208–226 (1988)
Cherrabi, A., Essannouni, H., Jabbouri, E., Ouadfel, A.: On a new extension of the zero-divisor graph. To appear in Algebra Colloquium. 2020 (arXiv:1806.11442)
Diestel, R.: Graph Theory. Springer-Verlag, New York (1997)
Hazewinkel, M., Gubareni, N., Kirichenko, V.V.: Algebras, Rings and Modules, vol. 1, Mathematics and its applications, Kluwer Academic Publishers (2005)
Huckaba, J.: Commutative Rings with Zero Divisors. Dekker, New York (1988)
Lucas, T.G.: The diameter of zero divisor graph. J. Algebra 301, 174–193 (2006)
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We would like to thank the referee for careful reading and helpful comments.
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Cherrabi, A., Essannouni, H., Jabbouri, E. et al. On a new extension of the zero-divisor graph (II). Beitr Algebra Geom 62, 945–953 (2021). https://doi.org/10.1007/s13366-020-00559-8
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DOI: https://doi.org/10.1007/s13366-020-00559-8