Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.
📌 3 Key Takeaways
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Objective & Hypothesis:In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J).
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Key Findings & Data:In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number.
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Conclusion & Impact:Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Ideals · clique number · complete r-partite
Abstract
In this paper, we introduce and investigate a new graph of a commutative ring R, denoted by G(R), with all nontrivial ideals of R as vertices, and two distinct vertices I and J are adjacent if and only if ann(I∩J)=ann(I)+ann(J). In this article, the basic properties and possible structures of the graph G(R) are studied and investigated as diameter, girth, clique number, cut vertex and domination number. We characterize all rings R for which G(R) is planar, complete and complete r-partite. We show that, if (R,M) is a local Artinian ring, then G(R) is complete if and only if Soc(R) is simple. Also, it is shown that if R is a ring with G(R) is r-regular, then either G(R) is complete or null graph. Moreover, we show that if R is an Artinian ring, then R is a serial ring if and only if G(R/I) is complete for each ideal I of R.
KHORAMDEL, M., ATANİ, S. E., PİSHHESARİ, S. D. (2019). On a graph of ideals of a commutative ring. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, Cilt: 68 - Sayı: 2, 2283–2297. https://ajindex.com/articles/747010
Akbari, S., Habibi, M., Majidinya, A., and Manaviyat, R., The inclusion ideal graph of rings, Communications in Algebra, 43 (2015), 2457--2465. Anderson, D. F., and Badawi, A., The total graph of a commutative ring, J. Algebra, 320 (2008), no. 7, 2706--2719. Anderson, D. F., and Badawi, A., Von Neumann Regular and Related Elements in Commutative Rings, Algebra Colloquium, (2012), Vol. 19, No. spec01: pp. 1017-1040.
Anderson, D. F., and Livingston, P. S., The zero-divisor graph of a commutative rings, J. Algebra, 217 (1999), 434--447. Atiyah, M. F., and Macdonald, I. G., Introduction to Commutative Algebra. Addison- Wesley Publishing Company, 1969.
Beck, I., Coloring of commutative rings. J. Algebra, 116 (1988), 208--226. Bondy, J. A., Murty, U. S. R., Graph Theory, Graduate Texts in Mathematics, 244. Springer, New York, 2008.
Bruns, W., Herzog, J., Cohen-Macaulay Rings, Cambridge University Press, 1997.
Camillo, V., Nicholson, W. K., Yousif, M. F., Ikeda-Nakayama Rings, Journal of Algebra, 226 (2000), 1001--1010. Chakrabarty, I., Ghosh, S., Mukherjee T. K., and Sen, M. K., Intersection graphs of ideals of rings, Discrete Math., 309 (2009) 5381--5392. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., The identity-summand graph of commutative semirings, J. Korean Math. Soc., 51 (2014), No. 1, pp. 189--202. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., Total graph of a commutative semiring with respect to identity-summand elements, J. Korean Math. Soc., 51 (3) (2014), 593--607. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., Total identity-summand graph of commutative semirings with respect to a co-ideal, J. Korean Math. Soc., 52(1) (2015), 159--176. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., A co-ideal based identity-summand graph of a commutative semiring, Commentationes Mathematicae Universitatis Carolinae, 56,3 (2015) 269--285. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., A graph associated to proper non-small ideals of a commutative ring, Comment.Math.Univ.Carolin., 58,1 (2017) 1--12. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., Ebrahimi Sarvandi, Z., Intersection graphs of co-ideals of semirings, Communications Faculty of Sciences University of Ankara Series A1: Mathematics and Statistics, Volume 68, Number 1, (2019), Pages 840--851. Ebrahimi Atani, S., Dolati Pish Hesari, S., Khoramdel, M., and Sedghi Shanbeh Bazari, M., Total graph of a 0-distributive lattice, Categories an