Classification of rings with toroidal annihilating-ideal graph

Document Type : Original paper

Authors

1 Department of Mathematics Manonmaniam Sundaranar University Tirunelveli

2 Manonmaniam Sundaranar University

Abstract

Let $R$ be a non-domain commutative ring with identity and $A^*(R)$ be the set of non-zero ideals with non-zero annihilators. We call an ideal $I$ of $R$, an annihilating-ideal if there exists a non-zero ideal $J$ of $R$ such that $IJ =(0)$. The annihilating-ideal graph of $R$ is defined as the graph $AG(R)$ with the vertex set $A^*(R)$ and two distinct vertices $I$ and $J$ are adjacent if and only if $IJ =(0)$. In this paper, we characterize all commutative Artinian nonlocal rings $R$ for which $AG(R)$ has genus one. 

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References

[1] G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaveisi, On the coloring of the annihilating-ideal graph of a commutative ring, Discrete Math. 312 (2012), no. 17, 2620–2626.
[2] G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaveisi, Minimal prime ideals and cycles in annihilating-ideal graphs, Rocky Mountain J. Math. 43 (2013), no. 5, 1415–1425.
[3] M. Afkhami and K. Khashyarmanesh, The cozero-divisor graph of a commutative ring., Southeast Asian Bull. Math. 35 (2011), no. 5, 753–762.
[4] S. Akbari, A. Alilou, J. Amjadi, and S.M. Sheikholeslami, The co-annihilatingideal graphs of commutative rings, Canad. Math. Bull. 60 (2017), no. 1, 3–11.
[5] D.F. Anderson, M. Axtell, and J. Stickles, Zero-divisor graphs in commutative rings: A survey, Commutative Algebra: Noetherian and Non-Noetherian Perspectives, (M. Fontana, S. E. Kabbaj, B. Olberding, I. Swanson) Springer-Verlag,
New York, Springer-Verlag, New York, 2011, pp. 23–45.
[6] D.F. Anderson and A. Badawi, The zero-divisor graph of a commutative semigroup: a survey, Groups, Modules, and Model Theory-Surveys and Recent Developments, Springer, 2017, pp. 23–39.
[7] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
[8] D. Archdeacon, Topological graph theory: a survey, Congr. Number. 115 (1996), 5 – 54.
[9] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42 (2014), no. 1, 108–121.
[10] , Recent results on the annihilator graph of a commutative ring: A survey, Nearrings, Nearfields and Related Topics, edited by K. Prasad et al., World Scientific, 2017, pp. 170–184.
[11] I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208–226.
[12] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Algebra Appl. 10 (2011), no. 4, 727–739.
[13] , The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 741 – 753.
[14] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications, vol. 290, American Elsevier, New York, 1976.
[15] T.T. Chelvam and K. Selvakumar, Central sets in the annihilating-ideal graph of commutative rings, J. Combin. Math. Combin. Comput. 88 (2014), 277–288.
[16] H.-J. Chiang-Hsieh, Classification of rings with projective zero-divisor graphs, J. Algebra 319 (2008), no. 7, 2789–2802.
[17] H.R. Maimani, M. Salimi, A. Sattari, and S. Yassemi, Comaximal graph of commutative rings, J. Algebra 319 (2008), no. 4, 1801–1808.
[18] W.S. Massey, Algebraic topology: an introduction, Harcourt, Brace & World, Inc., New York, 1967.
[19] K. Selvakumar and P. Subbulakshmi, On the crosscap of the annihilating-ideal graph of a commutative ring, Palestine J. Math. 7 (2018), no. 1, 151–160.
[20] K. Selvakumar, P. Subbulakshmi, and J. Amjadi, On the genus of the graph associated to a commutative ring, Discrete Math. Algorithms Appl. 9 (2017), no. 5, ID: 1750058 (11 pages).
[21] K. Wagner, U¨ ber eine erweiterung des satzes von kuratowski, Deutsche Math. 2 (1937), 280–285.
[22] H.-J. Wang, Graphs associated to co-maximal ideals of commutative rings, J. Algebra 320 (2008), no. 7, 2917–2933.
[23] A. T. White, Graphs, Groups and Surfaces, North-Holland, Amsterdam, 1973. 
Communications in Combinatorics and Optimization
Volume 3, Issue 2
December 2018
Pages 93-119

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