THE ANNIHILATING-IDEAL GRAPH OF A RING

Title & Authors
THE ANNIHILATING-IDEAL GRAPH OF A RING
ALINIAEIFARD, FARID; BEHBOODI, MAHMOOD; LI, YUANLIN;

Abstract
Let S be a semigroup with 0 and R be a ring with 1. We extend the definition of the zero-divisor graphs of commutative semigroups to not necessarily commutative semigroups. We define an annihilating-ideal graph of a ring as a special type of zero-divisor graph of a semigroup. We introduce two ways to define the zero-divisor graphs of semigroups. The first definition gives a directed graph $\small{{\Gamma}}$(S), and the other definition yields an undirected graph $\small{{\overline{\Gamma}}}$(S). It is shown that $\small{{\Gamma}}$(S) is not necessarily connected, but $\small{{\overline{\Gamma}}}$(S) is always connected and diam$\small{({\overline{\Gamma}}(S)){\leq}3}$. For a ring R define a directed graph $\small{{\mathbb{APOG}}(R)}$ to be equal to $\small{{\Gamma}({\mathbb{IPO}}(R))}$, where $\small{{\mathbb{IPO}}(R)}$ is a semigroup consisting of all products of two one-sided ideals of R, and define an undirected graph $\small{{\overline{\mathbb{APOG}}}(R)}$ to be equal to $\small{{\overline{\Gamma}}({\mathbb{IPO}}(R))}$. We show that R is an Artinian (resp., Noetherian) ring if and only if $\small{{\mathbb{APOG}}(R)}$ has DCC (resp., ACC) on some special subset of its vertices. Also, it is shown that $\small{{\overline{\mathbb{APOG}}}(R)}$ is a complete graph if and only if either $\small{(D(R))^2=0,R}$ is a direct product of two division rings, or R is a local ring with maximal ideal m such that $\small{{\mathbb{IPO}}(R)=\{0,m,m^2,R\}}$. Finally, we investigate the diameter and the girth of square matrix rings over commutative rings $\small{M_{n{\times}n}(R)}$ where $\small{n{\geq} 2}$.
Keywords
rings;semigroups;zero-divisor graphs;annihilating-ideal graphs;
Language
English
Cited by
References
1.
G. Aalipour, S. Akbari, M. Behboodi, R. Nikandish, M. J. Nikmehr, and F. Shahsavari, The classification of the annihilating-ideal graphs of commutative rings, Algebra Colloq. 21 (2014), no. 2, 249-256.

2.
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.

3.
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.

4.
S. Akbari and M. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847-855.

5.
S. Akbari and M. Mohammadian, Zero-divisor graphs of non-commutative rings, J. Algebra 269 (2006), no. 2, 462-479.

6.
S. Akbari and M. Mohammadian, On zero-divisor graphs of finite rings, J. Algebra 314 (2007), no. 1, 168-184.

7.
F. Aliniaeifard and M. Behboodi, Rings whose annihilating-ideal graphs have positive genus, J. Algebra Appl. 11 (2012), no. 3, 1250049, 13 pages.

8.
F. Aliniaeifard and M. Behboodi, Commutative rings whose zero-divisor graphs have positive genus, Comm. Algebra 41 (2013), no. 10, 3629-3634.

9.
D. F. Anderson, R. Levy, and J. Shapiro, Zero-divisor graphs, von Neumann regular rings, and Boolean algebras, J. Pure Appl. Algebra 180 (2003), no. 3, 221-241.

10.
D. F. Anderson and P. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434-447.

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.
M. Behboodi and Z. Rakeei, The annihilating-ideal graph of commutative rings II, J. Algebra Appl. 10 (2011), no. 4, 741-753.

14.
F. DeMeyer and L. DeMeyer, Zero-divisor graphs of semigroups, J. Algebra 283 (2005), no. 1, 190-198.

15.
F. DeMeyer, T. McKenzie, and K. Schneider, The Zero-divisor graph of a commutative semigroup, Semigroup Forum 65 (2002), no. 2, 206-214.

16.
F. DeMeyer and K. Schneider, Automorphisms and zero-divisor graphs of commutative rings, Commutative rings, 25-37, Nova Sci. Publ., Hauppauge, NY, 2002.

17.
N. S. Karamzadeh and O. A. S. Karamzadeh, On Artinian modules over Duo rings, Comm. Algebra 38 (2010), no. 9, 3521-3531.

18.
T. Y. Lam, A First Course in Non-Commutative Rings, Springer-Verlag, New York, 1991.

19.
S. B. Mulay, Cycles and symmetries of zero-divisors, Comm. Algebra 30 (2002), no. 7, 3533-3558.

20.
S. Redmond, The zero-divisor graph of a non-commutative ring, Commutative rings, 39-47, Nova Sci. Publ., Hauppauge, NY, 2002.

21.
S. Redmond, Structure in the zero-divisor graph of a noncommutative ring, Houston J. Math. 30 (2004), no. 2, 345-355.

22.
T. S. Wu, On directed zero-divisor graphs of finite rings, Discrete Math. 296 (2005), no. 1, 73-86.

23.
T. S. Wu, Q. Liu, and L. Chen, Zero-divisor semigroups and refinements of a star graph, Discrete Math. 309 (2009), no. 8, 2510-2518.

24.
T. S. Wu and D. C. Lu, Zero-divisor semigroups and some simple graphs, Comm. Al-gebra 34 (2006), no. 8, 3043-3052.

25.
T. S. Wu and D. C. Lu, Sub-semigroups determined by the zero-divisor graph, Discrete Math. 308 (2008), no. 22, 5122-5135.