When does the complement of the annihilating-ideal graph of a commutative ring admit a cut vertex?

Document Type : Research Paper

Authors

Saurashtra University, Rajkot, India

Abstract

 The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let $R$ be a ring. Let $\mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $\mathbb{A}(R)^{*} = \mathbb{A}(R)\backslash \{(0)\}$. The annihilating-ideal graph of $R$, denoted by $\mathbb{AG}(R)$  is an undirected simple graph whose vertex set is $\mathbb{A}(R)^{*}$ and distinct vertices $I, J$ are joined by an edge in this graph if and only if $IJ = (0)$. The aim of this article is to classify rings  $R$ such that $(\mathbb{AG}(R))^{c}$ ( that is,  the complement of $\mathbb{AG}(R)$)   is connected and admits a cut vertex.

Keywords

References

[1] G Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, The classi cation of
annihilating-ideal graph of commutative rings, Alg. Colloquium, 21, 249 (2014), doi:10.1143/S1005386714000200.
[2] G. Aalipour, S. Akbari, R. Nikandish, M.J. Nikmehr, and F. Shaiveisi, On the coloring of the annihilating-
ideal graph of a commutative ring, Discrete Math., 312 (2012), 2620-2625.
[3] .D.F. Anderson, M.C. Axtell, J.A. Stickles Zero-divisor graphs in commutative rings, in Commutative
Algebra, Noetherian and Non-Noetherian perspectives, M. Fontana, S.E. Kabbaj, B. Olberding, and I.
Swanson (Editors), Springer-Verlag, New York, 2011, 23-45.
[4] D.F. Anderson and P.S. Livingston, The zero-divisor graph of a commutative ring, J. Alg. 217 (1999),
434-447.
[5] M.F. Atiyah and I.G. Macdonald, Introduction to commutative algebra, Addison-Wesley, Reading, Mas-
sachusetts, 1969.
[6] M.C. Axtell, N. Baeth, and J.A. Stickles, Cut vertices in zero-divisor graphs of nite commutative rings,
Comm. Alg., 39(6) (2011), 2179-2188.
[7] R. Balakrishnan and K. Ranganathan, A Textbook of Graph Theory, Universitext, Springer, New york,
2000.
[8] I. . Beck, Coloring of commutative rings, J. Alg. 116 (1988), 208-226.
[9] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings I, J. Alg. Appl. 10 (2011),
727-739.
[10] M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Alg. Appl.10 (2011),
741-753.
[11] B. Cotee, C. Ewing, M. Huhn, C.M. Plaut, and E.D. Weber, Cut-Sets in zero-divisor graphs of nite
commutative rings, Comm. Alg. 39(8) (2011), 2849-2861.
[12] R. Gilmer and W. Heinzer, The Laskerian property, power series rings and Noetherian spectra, Proc. Amer.
Math. Soc. 79(1) (1980), 13-16.
[13] W. Heinzer and J. Ohm, Locally Noetherian commutative rings, Trans. Amer. Math. Soc. 158(2) (1971),
273-284.
[14] W. Heinzer and J. Ohm, On the Noetherian-like rings of E.G. Evans, Proc. Amer. Math. Soc. 34(1) (1972),
73-74.
[15] M. Hadian, Unit action and geometric zero-divisor ideal graph, Comm. Alg. 40 (2012), 2920-2930.
[16] I. Kaplansky, Commutative Rings, The University of Chicago Press, Chicago, 1974.
[17] T. Tamizh Chelvam and K. Selvakumar, On the connectivity of the annihilating-ideal graphs, Discuss.
Math. Gen. Alg. Appl. 35 (2015), 195-204.
[18] S. Visweswaran, Some results on the complement of the zero-divisor graph of a commutative ring, J. Alg.
Appl. 10(3) (2011), 573-595.
[19] S. Visweswaran, Some properties of the complement of the zero-divisor graph of a commutative ring, ISRN Alg. 2011 (2011), Article ID 591041, 24 pages.
[20] S. Visweswaran and Hiren D. Patel, Some results on the complement of the annihilating ideal graph of a
commutative ring, J. Algebra Appl. 14 (2015), doi: 10.1142/S0219498815500991, 23 pages.
[21] S. Visweswaran, When does the complement of the zero-divisor graph of a commutative ring admit a cut
vertex?, Palestine J. Math. 1(2) (2012), 138-147.
Algebraic Structures and Their Applications
Volume 2, Issue 2
November 2015
Pages 9-22

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