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Keywords:
idempotent graph; weak perfect graph; zero-divisor graph
idempotent graph; weak perfect graph; zero-divisor graph
Summary:
Let $R$ be a ring with nonzero identity. A graph $G_{{\rm Id}}(R)$ of $R$ with respect to idempotents of $R$ has elements of $R$ as vertices and distinct vertices $x$, $y$ are adjacent if and only if $x + y$ is an idempotent of $R$. In this paper, we prove that $G_{{\rm Id}}(R)$ is weakly perfect and provide a condition for the perfectness of the same. Further, we characterize finite abelian rings for which the complement of $G_{{\rm Id}}(R)$ is connected.
References:
[1] Abian, A.: Direct product decomposition of commutative semisimple rings. Proc. Am. Math. Soc. 24 (1970), 502-507. DOI 10.1090/S0002-9939-1970-0258815-X | MR 0258815 | Zbl 0207.05002
[2] Akbari, S., Habibi, M., Majidinya, A., Manaviyat, R.: On the idempotent graph of a ring. J. Algebra Appl. 12 (2013), Article ID 1350003, 14 pages. DOI 10.1142/S0219498813500035 | MR 3063442 | Zbl 1266.05051
[3] Anderson, D. F., Badawi, A.: Von Neumann regular and related elements in commutative rings. Algebra Colloq. 19 (2012), 1017-1040. DOI 10.1142/S1005386712000831 | MR 3073393 | Zbl 1294.13029
[4] Anderson, D. F., LaGrange, J. D.: Abian's poset and the ordered monoid of annihilator classes in a reduced commutative rings. J. Algebra Appl. 13 (2014), Article ID 1450070, 18 pages. DOI 10.1142/S0219498814500704 | MR 3225137 | Zbl 1317.06020
[5] Anderson, D. D., Naseer, M.: Beck's coloring of a commutative ring. J. Algebra 159 (1993), 500-514. DOI 10.1006/jabr.1993.1171 | MR 1231228 | Zbl 0798.05067
[6] Beck, I.: Coloring of commutative rings. J. Algebra 116 (1988), 208-226. DOI 10.1016/0021-8693(88)90202-5 | MR 0944156 | Zbl 0654.13001
[7] Chudnovsky, M., Robertson, N., Seymour, P., Thomas, R.: The strong perfect graph theorem. Ann. Math. (2) 164 (2006), 51-229. DOI 10.4007/annals.2006.164.51 | MR 2233847 | Zbl 1112.05042
[8] Cvetko-Vah, K., Dolžan, D.: Indecomposability graphs of rings. Bull. Aust. Math. Soc. 77 (2008), 151-159. DOI 10.1017/S0004972708000154 | MR 2411874 | Zbl 1194.16034
[9] Han, J., Park, S.: Additive set of idempotents in rings. Commun. Algebra 40 (2012), 3551-3557. DOI 10.1080/00927872.2011.591862 | MR 2981152 | Zbl 1256.16024
[10] Patil, A., Momale, P. S.: Idempotent graphs, weak perfectness, and zero-divisor graphs. Soft Comput. 25 (2021), 10083-10088. DOI 10.1007/s00500-021-05982-0 | Zbl 1498.05128
[11] Patil, A., Waphare, B. N., Joshi, V.: Perfect zero-divisor graphs. Discrete Math. 340 (2017), 740-745. DOI 10.1016/j.disc.2016.11.027 | MR 3603554 | Zbl 1355.05115
[12] Razaghi, S., Sahebi, S.: A graph with respect to idempotents of a ring. J. Algebra Appl. 20 (2021), Article ID 2150105, 8 pages \99999DOI99999 10.1142/S021949882150105X . DOI 10.1142/S021949882150105X | MR 4256354 | Zbl 1496.16037
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