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 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 AGP(R). It is a (undirected) graph with vertices AP(R)=A(R)∩P(R)∖{(0)}, where   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 AGP(R). For instance, we characterize rings for which AGP(R) is finite graph, complete graph, bipartite graph or star graph. Also, we study diameter and girth of AGP(R). Finally, we compare  the principal ideal subgraph AGP(R) and spectrum subgraph AGs(R).

Keywords

References

[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].
Algebraic Structures and Their Applications
Volume 3, Issue 1
February 2016
Pages 39-52

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