On the Genus of the Co-Annihilating Graph of Commutative Rings

  • 1 Department of Mathematics, 627 012, Tirunelveli, India

Abstract

Let R be a commutative ring with identity and 𝒰R be the set of all nonzero non-units of R. The co-annihilating graph of R, denoted by π’žπ’œR, is a graph with vertex set 𝒰R and two vertices x and y are adjacent whenever ann(x) ∩ ann(y) = (0). In this paper, we characterize all commutative Artinian non-local rings R for which the π’žπ’œR has genus one and two. Also we characterize all commutative Artinian non-local rings R for which π’žπ’œR has crosscap one. Finally, we characterize all finite commutative non-local rings for which g(Π“2(R)) = g(π’žπ’œR) = 0 or 1.

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  • [1] S. Akbari, H.R. Maimani and S. Yassemi, When a zero-divisor graph is planar or a complete r-partite graph, J. Algebra 270 (2003) 169–180. doi:10.1016/S0021-8693(03)00370-3

  • [2] J. Amjadi and A. Alilou, The co-annihilating graph of a commutative ring, Discrete Math. Alg. and Appl. 10 (1) (2018) 1850013(1–12). doi:10.1142/S1793830918500131

  • [3] S. Akbari, J. Amjadi, A. Alilou and S.M. Sheikholeslami, The co-annihilating ideal graphs of a commutative ring, Canad. Math. Bull. 60 (2017) 3–11. doi:10.4153/CMB-2016-017-1

  • [4] S. Akbari, M. Habibi, A. Majidinya and R. Manaviyat, A note on comaximal graph of non-commutative Rings, Algebr. Reprsent. Theory 16 (2013) 303–307. doi:10.1007/s10468-011-9309-z

  • [5] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999) 434–447. doi:10.1006/jabr.1998.7840

  • [6] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra (Addison-Wesley Publishing Company, Londan, 1969).

  • [7] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42 (2014) 108–121. doi:10.1080/00927872.2012.707262

  • [8] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings-I, J. Algebra Appl. 10 (2011) 741–753. doi:10.1142/S0219498811004902

  • [9] R. Belshoff and J. Chapman, Planar zero-divisor graphs, J. Algebra 316 (2007) 471–480. doi:10.1016/j.jalgebra.2007.01.049

  • [10] G. Chartrand and P. Zhang, A First Course in Graph Theory (Dover Publications, Mineola, NY, USA, 2012).

  • [11] Hung-Jen and Chiang-Hsieh, Classification of rings with projective zero-divisor graphs, J. Algebra 319 (2008) 2789–2802. doi:10.1016/j.jalgebra.2007.10.015

  • [12] M. Bojan and T. Carsten, Graphs on Surfaces (John Hopkins University Press, Baltimore, 2001).

  • [13] N. Bloomfield and C. Wickham, Classification of rings with genus two zero-divisor graphs, Comm. Algebra 33 (2010) 2965–2980. doi:10.1080/00927870903100093

  • [14] K. Selvakumar and M. Subajini, Crosscap of the non-cyclic graph of groups, AKCE Inter. J. Graphs and Combin. 13 (2016) 235–240. doi:10.1016/j.akcej.2016.06.013

  • [15] H.J. Wang, Zero-divisor graphs of genus one, J. Algebra 304 (2006) 666–678. doi:10.1016/j.jalgebra.2006.01.057

  • [16] H.J. Wang, Graphs associated to co-maximal ideals of commutative rings, J. Algebra 320 (2008) 2917–2933. doi:10.1016/j.jalgebra.2008.04.013

  • [17] H.J. Wang, Co-maximal graph of non-commutative rings, Linear Algebra Appl. 430 (2009) 633–641. doi:10.1016/j.laa.2008.09.022

  • [18] A.T. White, Graphs, Groups and Surfaces (North-Holland Publishing Company, Amsterdam, 1973).

  • [19] C. Wickham, Classification of rings with genus one zero-divisor graphs, Comm. Algebra 36 (2008) 325–345. doi:10.1080/00927870701713089

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