# The principal small intersection graph of a commutative ring

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https://doi.org/10.33044/revuma.3486## Abstract

Let $R$ be a commutative ring with non-zero identity. The small intersection graph of $R$, denoted by $G(R)$, is a graph with the vertex set $V(G(R))$, where $V(G(R))$ is the set of all proper non-small ideals of $R$ and two distinct vertices $I$ and $J$ are adjacent if and only if $I \cap J$ is not small in $R$. In this paper, we introduce a certain subgraph $PG(R)$ of $G(R)$, called the principal small intersection graph of $R$. It is the subgraph of $G(R)$ induced by the set of all proper principal non-small ideals of $R$. We study the diameter, the girth, the clique number, the independence number and the domination number of $PG(R)$. Moreover, we present some results on the complement of the principal small intersection graph.

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## References

S. Akbari and S. Khojasteh, Commutative rings whose cozero-divisor graphs are unicyclic or of bounded degree, *Comm. Algebra* **42** no. 4 (2014), 1594–1605. DOI MR Zbl

S. Akbari, H. A. Tavallaee, and S. Khalashi Ghezelahmad, Intersection graph of submodules of a module, *J. Algebra Appl.* **11** no. 1 (2012), Paper no. 1250019, 8 pp. DOI MR Zbl

D. F. Anderson and A. Badawi, The total graph of a commutative ring, *J. Algebra* **320** no. 7 (2008), 2706–2719. DOI MR Zbl

D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, *J. Algebra* **217** no. 2 (1999), 434–447. DOI MR Zbl

S. E. Atani, S. D. Pish Hesari, and M. Khoramdel, A graph associated to proper non-small ideals of a commutative ring, *Comment. Math. Univ. Carolin.* **58** no. 1 (2017), 1–12. DOI MR Zbl

M. F. Atiyah and I. G. Macdonald, *Introduction to commutative algebra*, Addison-Wesley, Reading, Mass.-London-Don Mills, Ont., 1969. MR

I. Chakrabarty, S. Ghosh, T. K. Mukherjee, and M. K. Sen, Intersection graphs of ideals of rings, *Discrete Math.* **309** no. 17 (2009), 5381–5392. DOI MR Zbl

B. Csákány and G. Pollák, The graph of subgroups of a finite group (Russian), *Czechoslovak Math. J.* **19 (94)** no. 2 (1969), 241–247. DOI MR Zbl

F. Heydari, The $M$-intersection graph of ideals of a commutative ring, *Discrete Math. Algorithms Appl.* **10** no. 3 (2018), Paper no. 1850038, 11 pp. DOI MR Zbl

S. Khojasteh, The intersection graph of ideals of $ℤ_m$, *Discrete Math. Algorithms Appl.* **11** no. 4 (2019), Paper no. 1950037, 12 pp. DOI MR Zbl

S. Khojasteh, The complement of the intersection graph of ideals of a poset, *J. Algebra Appl.* **22** no. 11 (2023), Paper no. 2350236, 13 pp. DOI MR Zbl

R. Y. Sharp, *Steps in commutative algebra*, London Mathematical Society Student Texts 19, Cambridge University Press, Cambridge, 1990. MR Zbl

R. Wisbauer, *Foundations of module and ring theory*, Algebra, Logic and Applications 3, Gordon and Breach Science Publishers, Philadelphia, PA, 1991. MR Zbl

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