Original Research Article

Article volume = 2023 and issue = 2

Pages: 140–148

Article publication Date: November 25, 2023

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# Semisimple-Intersection Graphs of Ideals of Rings

#### Ahmed H. Alwan

Department of Mathematics, College of Education for Pure Sciences, University of Thi-Qar, Thi-Qar, Iraq

##### Abstract:

In this paper, a new kind of graph on a ring is introduced and investigated. Let R be a ring with unity. Semisimpleintersection graph of a ring R, denoted by GS(R), is an undirected simple graph with all nonzero ideals of R as vertices and two distinct vertices I and J are adjacent if and only if I ∩ J is a nonzero semisimple ideal of R. In this article, we investigate the basic properties of these graphs to relate the combinatorial properties of GS(R) to the algebraic properties of the ring R. We determine the diameter and the girth of GS(R). We obtain some results for connectedness and bipartiteness of these graphs, as well as give a formula to compute the clique numbers of GS(R). We observed that the graph GS(R) is complete if and only if every proper ideal of R either simple or semisimple and every pair of ideals in R have non-zero intersection.

##### Keywords:

Semisimple-intersection graph; rings; ideals; clique number; girth; diameter.

##### References:
• [1] S. Akbari, R. Nikandish, Some results on the intersection graph of ideals of matrix algebras, Linear Multilinear A., 62 (2014), 195-206. 1
• [2] A. H. Alwan, Maximal submodule graph of a module, J. Discrete Math. Sci. Cryptogr., 24(7), 2021, 1941-1949. 1
• [3] A. H. Alwan, Maximal ideal graph of commutative semirings, Int. J. Nonlinear Anal. Appl., 12(1), 2021, 913-926. 1
• [4] A. H. Alwan, Small intersection graph of subsemimodules of a semimodule. Commun. Combin., Cryptogr. & Computer Sci., 2022; 1, 15-22. 1
• [5] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), 208-226. 1
• [6] J. A. Bondy, U. S. R. Murty, Graph theory, Springer-Verlag, London, 2011. 2
• [7] J. Bosak, The graphs of semigroups, in Theory of Graphs and its Applications, (Academic Press, New York, 1964), pp. 119-125. 1
• [8] I. Chakrabarty, S. Ghosh, T. K. Mukherjee, M. K. Sen, Intersection graphs of ideals of rings, Discrete Math., 309 (2009), 5381-5392. 1
• [9] I. Chakrabarty, J. V. Kureethara, A survey on the intersection graphs of ideals of rings, Commun. Combin. Optim., 7 (2022), 121-167. 1
• [10] P. M. Cohn, Introduction to ring theory, British Library Cataloguing in Publication Data, Springer-Verlag, London Berlin Heidelberg, 2000. 2
• [11] J. Matczuk, M. Nowakowska, E. R Puczy lowski, Intersection graphs of modules and rings, J. Algebra Appl., 17 (2018), 185-131. 1
• [12] F. Moh’d, M. Ahmed, Simple-intersection graphs of rings, AIMS Math., 8(1), 2022, 1040-1054. 1, 4.12
• [13] E. A. Osba, The intersection graph for finite commutative principal ideal rings, Acta Math. Acad. Paedagog. Nyh´azi., 32 (2016), 15-22. 1
• [14] N. J. Rad, S. H. Jafari, S. Ghosh, On the intersection graphs of ideals of direct product of rings, J. Discuss. Math.-Gen. Algebra Appl., 34 (2014), 191-201. 1