On the girth of the annihilating-ideal graph of a commutative ring

Subject Areas : History and biography

M.  Ahrari ¹ (Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran)
Sh. A.  Safari Sabet ² (Department of Mathematics, Islamic Azad University, Central Tehran Branch, Tehran, Iran)
B.  Amini ³ (Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Iran)

Received: 2015-04-05 Accepted : 2015-12-13 Published : 2015-08-01

Keywords: annihilating-ideal graph, star graph, bipartite graph, girth,

Abstract :

The annihilating-ideal graph of a commutative ring $R$ isdenoted by $AG(R)$, whose vertices are all nonzero ideals of $R$with nonzero annihilators and two distinct vertices $I$ and $J$are adjacent if and only if $IJ=0$. In this article, we completelycharacterize rings $R$ when $gr(AG(R))\neq 3$.

References:

[1] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr and F. Shaveisi, The classification of annihilating-ideal graphs of commutative rings, Algebra Colloquium 21(2) (2014) 249-256.

[2] A. Amini, B. Amini, E. Momtahan and M. H. Shirdareh Haghighi, On a graph of ideals, Acta Math. Hungar 134 (3) (2012) 369–384.

[3] D. F. Anderson, M. C. Axtell and J. A. Stickles, Zero-divisor graphs in commutative rings, in Commutative Algebra, Noetherian and Non-Noetherian Perspective, 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(7) (2008) 2706-2719.

[5] N. Ashrafi, H. R. Maimani, M. R. Pouranki and S. Yassemi, Unit graphs associated with rings, Comm. Algebra 38 (2010) 2851-2871.

[6] M. Baziar, E. Momtahan and S. Safaeeyan, A zero-divisor graph for modules with respect to their (first) dual, J. Algebra Appl. 12(2) (2013) 1250151.

[7] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988) 208–226.

[8] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10(4) (2011) 727-739.

[9] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10(4) (2011) 741-753.

[10] S. P. Redmond, The zero-divisor graph of a non-commutative ring, Int. J. Commut. Rings, 1(4) (2002) 203-211.