The principal ideal subgraph of the annihilating-ideal graph of commutative rings

Document Type : Research Paper

Authors

Islamic Azad University, Science and Research Branch, Tehran, Iran

Abstract

Let $R$ be a commutative ring with identity and $\mathbb{A}(R)$ be the set   of ideals of $R$ with non-zero annihilators. In this paper, we first introduce and investigate the principal ideal subgraph of the annihilating-ideal graph of $R$, denoted by $\mathbb{AG}_P(R)$. It is a (undirected) graph with vertices $\mathbb{A}_P(R)=\mathbb{A}(R)\cap \mathbb{P}(R)\setminus \{(0)\}$, where   $\mathbb{P}(R)$ is the set of  proper principal ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ=(0)$. Then, we study some basic properties of $\mathbb{AG}_P(R)$. For instance, we characterize rings for which $\mathbb{AG}_P(R)$ is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of $\mathbb{AG}_P(R)$. Finally, we compare  the principal ideal subgraph $\mathbb{AG}_P(R)$ and spectrum subgraph $\mathbb{AG}_s(R)$.

Keywords


[1] G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish and M. J. Nikmehr and F. Shahsavari, The classification of the annihilating-ideal graph of a commutative ring, Algebra Colloquium 21 (2014) 249-256.
[2] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shahsavari, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math. 312 (2012) 2620-2626.
[3] G. Aalipour, S. Akbari, R. Nikandish, M. J. Nikmehr and F. Shahsavari, Minimal prime ideals and cycles in
annihilating-ideal graphs, Rocky Mountain J. Math. 5 (2013) 1415-1425.
[4] F. Aliniaeifard and M. Behboodi, Rings whose annihilating-ideal graphs have positive genus, J. Algebra Appl. 11, 1250049 (2012) [13 pages] DOI: 10.1142/S0219498811005774.
[5] F. Aliniaeifard, M. Behboodi, E. Mehdi-Nezhad, and A.M. Rahimi, On the diameter and girth of an annihilating-ideal graph, to apear.
[6] F. Aliniaeifard, M. Behboodi, E. Mehdi-Nezhad and A. M. Rahimi, The annihilating-ideal graph of a commutative ring with respect to an ideal, Commun. Algebra 42 (2014) 2269-2284.
[7] D. F. Anderson and A. Badawi, The total graph of a commutative ring, J. Algebra 320 (2008) 2706-2719.
[8] D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434-447.
[9] L. Anderson, A First Course in Discrete Mathematice, Springer Undergraduate Mathematics Series,2000.
[10] M. Baziar, E. Momtahan and S. Safaeeyan, A zero-divisor graph for modules with respect to their (first) dual, J. Algebra Appl. 12, 1250151 (2013) [11 pages] DOI: 10.1142/S0219498812501514.
[11] M. Behboodi, Zero divisor graphs for modules over commutative rings, J. Commut. Algebra 4 (2012) 175-197.
[12] M. Behboodi, Z. Rakeei,The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011) 727-739.
[13] M. Behboodi, Z. Rakeei,The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011) 740-753.
[14] I. Chakrabarty, S. Ghosh, T. K. Mukherjee and M. K. Sen, Intersection graphs of ideals of rings, Discrete Math. 309 (2009), 5381-5392.
[15] W. K. Nicholson and E. s , anchez-Campos, Rings with the dual of the isomorphism theorem , J. Algebra 271 (1) (2004), 391-406.
[16] R. Nikandish and H. R. Maimani, Dominating sets of the annihilating-ideal graphs, Electronic Notes in Discrete Math. 45 (2014) 17-22.
[17] R. Y. Sharp, Steps in commutative algebra Cambridge University Press, Cambridge, 1991.
[18] R. Taheri, M. Behboodi and A. Tehranian, The spectrum subgraph of the annihilating-ideal graph of a commutative ring, J. Algebra Appl. 14 (2015) [19 page].