Some properties and domination number of the complement of a new graph associated to a commutative ring

Subject Areas : Statistics

J.  amjadi ¹ (Department of Mathematics ,Azarbaijan Shahid Madani University, Tabriz, I.R. Iran)

Received: 2017-06-29 Accepted : 2018-03-07 Published : 2018-01-01

Keywords: ایده‌آ‌ل پوچ شونده, ایده‌آ‌ل N- اول ماکسیمال, گراف مسطح, عدد احاطه‌ای,

Abstract :

In this paper some properties of the complement of a new graph associated with a commutative ring are investigated ....

References:

 

_{[1] M. Afkhami and, K. Khashyarmanesh, Some results on the annihilator graph of a commutative ring, Czechoslovak Mat. J. 67 (2017), 151-169.}

 

_{[2] M. Afkhami, N. Hoseini and K. Khashyarmanesh, The annihilator ideal graph of a commutative ring, Note Math. 36 (2017),1-10.}

 

_{[3] S. Akbari, A. Alilou, J. Amjadi and S.M. Sheikholeslami, The co-annihilating ideal graphs of commutative rings, Canad. Math. Bull. 60 (2017), 3-11.}

 

_{[4] A. Alilou, J. Amjadi and S.M. Sheikholeslami, A new graph associated to a commutative  ring,  Discrete  Math. Algorithm. Appl. 8 (2016), (13 pages).}

 

_{[5] A. Alilou and J. Amjadi, The sum-annihilating essential ideal graph of a commutative ring, Commun. Comb. Optim. 1 (2016), 117-135.}

 

_{[6] D. D. Anderson, and M. Naseer, Beck’s coloring of a commutative ring, J. Algebra 159 (1993), 500-514.}

 

_{[7] D.F. Anderson and A. Badawi, The zero-divisor graph of a commutative semigroup: A survey, Groups, Modules, and Model Theory-surveys and Developments (2017), 23-99.}

 

_{[8] D.F. Anderson and P. S. Livingston, The Zero-Divisor Graph of a Commutative Ring, J. Algebra 217 (1999), 434-447.}

 

_{[9] H.Ansari-Toroghy, F. Farshadifar and F. Mahboobi-Abkenar, On the ideal-based zero-divisor graph, IEJA (to appear).}

 

_{[10] H.Ansari-Toroghy, F. Farshadifar and F. Mahboobi-Abkenar, The small intersection graph relative to multiplication modules, Journal of Algebra and Related Topics  4 (2016), 21-32.}

 

_{[11] M. F. Atiyah, I.G. Macdonald, Introduction to Commutative Algebra, Addison-wesley publishing company (1969).}

 

_{[12] A. Badawi, Recent results on the annihilator graph of a commutative ring: A survey, Nearrings, Nearfields and Related Topics, Publisher: World Scientific (2016).}

 

_{[13] I.­ Beck, Coloring­ of­ a commutative ring, J. Algebra 116 (1988), 208-226.}

 

_{[14] M. Behboodi and Z. Rakeei, The Annihilating-ideal graph of commutative ring I, J. Algebra Appl. 10 (2011), 727-739.}

 

_{[15] M. Behboodi and Z. Rakeei, The Annihilating-ideal graph of commutative ring II, J. Algebra Appl. 10 (2011), 741-753.}

 

_{[16] S. Ebrahimi Atani, S. Dolati Pish Hesari and M. Khoramdel, A graph associated to proper non-small ideals of a commutative ring, Comment. Math. Univ. Carolin 58 (2017), 1-12.}

 

_{[17] Z. Ebrahimi Sarvandi and S. Ebrahimi Atani, The total graph of acommutative smiring with respect to proper ideals, J. Algebra and relative topics 3 (2015), 27-41.}

 

_{[18] R.Gilmer, Multiplicative Ideal}

_{Theory, Marcel-Dekker, New York, (1972).}

 

_{[19] T.W. Haynes, S. T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs, Marcel Dekker, Inc, New York (1998).}

 

_{[20] W. Heinzer and J. Ohm, On the Noetherian-like rings of E. G. Evans, Proc. Amer. Math. Soc. 34 (1972), 73-74.}

 

_{[21] J.A. Huckaba, Commutative  rings with zero divisors, Marcel Dekker, Inc, New York (1988).}

 

_{[22] I. Kaplansky, Commutative rings, The University of Chicago Press, Chicago (1974).}

 

_{[23] K. Kuratowski, Sur le Problèm des courbes gauches en topologie, Fund. Mat. 15 (1930), 271-283.}

 

_{[24] E. Matlis, The minimal prime spectrum of reduced ring, Illions J. Math. 27 (1983), 353-391.}

 

_{[25] R. Nikandish, M. J. Nikmehr and M. Bakhtyari, Coloring of the annihilator graph of a commutative ring, J. Algebra Appl. 15 (206), (13 pages).}

 

_{[26] R. Nikandish, M. J. Nikmehr and M. Bakhtyari, Some fiuther results on the annihilator ideal graph of a commutative ring, U.P.B.Sci. Bull., Series A 79 (2017), 99-108.}

 

_{[27] R.Y. Sharp, Steps in Commutative Algebra (second edition), London Mathematics Society Students Texts, 51 Cambridge University Press, Cambridge, (1990).}

 

_{[28] S. Visweswaran and Hiren D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithm. Appl. 6 (2014) [22 pages].}

 

[29] D.B. West, Introduction to Graph Theory (second edition), Prentice Hall, USA