The sum-annihilating essential ideal graph of a commutative ring

Document Type : Original paper

Authors

Azarbaijan Shahid Madani University

Abstract

Let $R$ be a commutative ring with identity. An ideal $I$ of a ring $R$ is called an annihilating ideal if there exists $r\in R\setminus \{0\}$ such that $Ir=(0)$ and an ideal $I$ of $R$ is called an essential ideal if $I$ has non-zero intersection with every other non-zero ideal of $R$. The sum-annihilating essential ideal graph of $R$, denoted by $\mathcal{AE}_R$, is a graph whose vertex set is the set of all non-zero annihilating ideals and two vertices $I$ and $J$ are adjacent whenever ${\rm Ann}(I)+{\rm Ann}(J)$ is an essential ideal. In this paper we initiate the study of the sum-annihilating essential ideal graph. We first characterize all rings whose sum-annihilating essential ideal graph are stars or complete graphs and then establish sharp bounds on domination number of this graph. Furthermore determine all isomorphism classes of Artinian rings whose sum-annihilating essential ideal graph has genus zero or 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] S. Akbari, A. Alilou, J. Amjadi, and S.M. Sheikholeslami, The coannihilating ideal graphs of commutative rings, Canadian Math. Bull. to appear.
[3] S. Akbari and F. Heydari, The regular graph of a commutative ring, Period. Math. Hungar. 67 (2013), no. 2, 211–220.
[4] S. Akbari and A. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 296 (2006), no. 2, 462–479.
[5] D.F. Anderson and A. Badawi, On the zero-divisor graph of a ring, Comm. Algebra 36 (2008), 3073–3092.
[6] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434–447.
[7] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison-Wesley Publishing Company, 1969.
[8] A. Badawi, On the annihilator graph of a commutative ring, Comm. Algebra 42 (2014), 108–121.
[9] I. Beck, Coloring of a commutative ring, J. Algebra 116 (1988), no. 1, 208–226.
[10] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings ii, J. Algebra Appl. 10 (2001), no. 4, 741–753.
[11] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings i, J. Algebra Appl. 10 (2011), no. 4, 727–739.
[12] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Domination in graphs: Advanced topics, Marcel Dekker, Inc., New York, 1998.
[13] T.W. Haynes, S.T. Hedetniemi, and P.J. Slater, Fundamentals of domination in graphs, Marcel Dekker, Inc., New
York, 1998.
[14] K. Kuratowski, Sur le probl`em des courbes gauches en topologie, Fund. Math. 15 (1930), 271–283.
[15] T.Y. Lam, A first course in non-commutative rings, Springer-Verlag (New York, Inc., 1991.
16] M. Matlis, The minimal prime spectrum of a reduced ring, Illinois J. Math. 27 (1983), no. 3, 353–391.
[17] R. Nikandish and H.R. Maimani, Dominating sets of the annihilating-ideal graphs, Electron. Notes Discrete Math. 45 (2014), 17–22.
[18] S.P. Redmond, The zero-divisor graph of a non-commutative ring, Internat. J. Commutative rings 1 (2002), 203–211.
[19] G. Ringel, Das gescblecht des vollstandingen paaren graphen, Abh. Math. Sem. Univ. Hamburg 28 (1965), 139–150.
[20] G. Ringel and J.W.T. Youngs, Solution of the heawood map-coloring problem, Proc. Nat. Acad. Sci. U.S.A. 60 (1968), 438–445.
[21] P.K. Sharma and S.M. Bhatwadekar, A note on graphical representation of rings, J. Algebra 176 (1995), no. 1, 124–127.
[22] A. Tehranian and H.R. Maimani, A study of the total graph, Iranian J. Math. Sci. Info. 6 (2011), no. 2, 75–80.
[23] S. Visweswaran and H.D. Patel, A graph associated with the set of all nonzero annihilating ideals of a commutative ring, Discrete Math. Algorithms Appl. 6 (2014), no. 4, 1450047 (22 pages).
[24] S. Visweswaran and H.D. Patel, When is the annihilating ideal graph of a zero-dimensional quasisemilocal commutative ring complemented?, Arab. J. Math. Sci. 22 (2016), no. 1, 1–21.
[25] D.B. West, Introduction to graph theory (second edition), Prentice Hall, USA, 2001.
Communications in Combinatorics and Optimization
Volume 1, Issue 2 - Serial Number 2
December 2016
Pages 117-135

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