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:

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##### Cite this article as:

- Ahmed H. Alwan, Semisimple-Intersection Graphs of Ideals of Rings, Communications in Combinatorics, Cryptography & Computer Science, 2023(2), PP.140–148, 2023
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