Document Type : Original paper

**Authors**

Saurashtra University

**Abstract**

The rings considered in this article are commutative with identity which admit at least one nonzero proper ideal. Let $R$ be a ring. We denote the collection of all ideals of $R$ by $\mathbb{I}(R)$ and $\mathbb{I}(R)\backslash \{(0)\}$ by $\mathbb{I}(R)^{*}$. Alilou et al. [A. Alilou, J. Amjadi and S.M. Sheikholeslami, {\em A new graph associated to a commutative ring}, Discrete Math. Algorithm. Appl. {\bf 8} (2016) Article ID: 1650029 (13 pages)] introduced and investigated a new graph associated to $R$, denoted by $\Omega_{R}^{*}$ which is an undirected graph whose vertex set is $\mathbb{I}(R)^{*}\backslash \{R\}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if either $(Ann_{R}I)J = (0)$ or $(Ann_{R}J)I = (0)$. Several interesting theorems were proved on $\Omega_{R}^{*}$ in the aforementioned paper and they illustrate the interplay between the graph-theoretic properties of $\Omega_{R}^{*}$ and the ring-theoretic properties of $R$. The aim of this article is to investigate some properties of $(\Omega_{R}^{*})^{c}$, the complement of the new graph $\Omega_{R}^{*}$ associated to $R$.

**Keywords**

**Main Subjects**

[1] G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math. 312 (2012), no. 17, 2620–2626.

[2] A. Alilou, J. Amjadi, and S.M. Sheikholeslami, A new graph associated to a commutative ring, Discrete Math. Algorithms Appl. 8 (2016), no. 2, Article ID: 1650029 (13 pages).

[3] D.F. Anderson, M.C. Axtell, and J.A. Stickles, Zero-divisor graphs in commutative rings, Commutative Algebra, Noetherian and Non-Noetherian Perspectives, eds. M. Fontana, S.E. Kabbaj, B. Olberding and I. Swanson (Spring-Verlag, New York) (2011), 23–45.

[4] D.F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008), no. 7, 2706–2719.

[5] M. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, AddisonWesley, Massachusetts, 1969.

[6] M. Axtell, J. Coykendall, and J.A. Stickles, Zero-divisor graphs of polynomials and power series over commutative rings, Comm. Algebra 33 (2005), no. 6, 2043–2050.

[7] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42 (2014), no. 1, 108–121.

[8] R. Balakrishnan and K. Ranganathan, A text book of graph theory, Springer Science & Business Media, 2000.

[9] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226.

[10] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727–739.

[11] , The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 741–753.

[12] R. Gilmer, Multiplicative Ideal Theory, vol. 12, Marcel-Dekker, New York, 1972.

[13] R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer. Math. Soc. 79 (1980), no. 1, 13–16.

[14] M. Hadian, Unit action and the geometric zero-divisor ideal graph, Comm. Algebra 40 (2012), no. 8, 2920–2931.

[15] W. Heinzer and J. Ohm, On the Noetherian-like rings of E.G. Evans, Proc. Amer. Math. Soc. 34 (1972), no. 1, 73–74.

[16] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.

[17] S. Visweswaran, Some results on the complement of the zero-divisor graph of a commutative ring, J. Algebra Appl. 10 (2011), no. 3, 573–595.

[18] S. Visweswaran and H.D. Patel, A graph associated with the set of all non-zero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 6 (2014), no. 4, Article ID: 1450047 (22 pages).

[19] S. Visweswaran and P. Sarman, On the complement of a graph associated with the set of all non-zero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 8 (2016), no. 3, Article ID: 1650043 (22 pages).

September 2017

Pages 119-138