THE ZERO-DIVISOR GRAPH UNDER A GROUP ACTION IN A COMMUTATIVE RING

Title & Authors
THE ZERO-DIVISOR GRAPH UNDER A GROUP ACTION IN A COMMUTATIVE RING
Han, Jun-Cheol;

Abstract
Let R be a commutative ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. We will investigate some ring theoretic properties of R by considering $\small{\Gamma}$(R), the zero-divisor graph of R, under the regular action on X by G as follows: (1) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then there is a vertex of $\small{\Gamma}$(R) which is adjacent to every other vertex in $\small{\Gamma}$(R) if and only if R is a local ring or $\small{R\;{\simeq}\;\mathbb{Z}_2\;{\times}\;F}$ where F is a field; (2) If R is a local ring such that X is a union of n distinct orbits under the regular action of G on X, then all ideals of R consist of {{0}, J, $\small{J^2}$, $\small{\ldots}$, $\small{J^n}$, R} where J is the Jacobson radical of R; (3) If R is a ring such that X is a union of a finite number of orbits under the regular action on X by G, then the number of all ideals is finite and is greater than equal to the number of orbits.
Keywords
zero-divisor graph;regular action;orbit;local ring;
Language
English
Cited by
References
1.
S. Akbari and A. Mohammadian, On the zero-divisor graph of a commutative ring, J. Algebra 274 (2004), no. 2, 847-855.

2.
D. F. Anderson, A. Frazier, A. Lauve, and P. S. Livingston, The zero-divisor graph of a commutative ring. II, Ideal theoretic methods in commutative algebra (Columbia, MO, 1999), 61-72, Lecture Notes in Pure and Appl. Math., 220, Dekker, New York, 2001.

3.
D. F. Anderson and P. S. Livingston, The zero-divisor graph of a commutative ring, J. Algebra 217 (1999), no. 2, 434-447.

4.
I. Beck, Coloring of commutative rings, J. Algebra 116 (1988), no. 1, 208-226.

5.
F. DeMeyer and L. DeMeyer, Zero divisor graphs of semigroups, J. Algebra 283 (2005), no. 1, 190-198.

6.
R. Diestel, Graph Theory, pringer-Verlag, New York, 1997.

7.
J. Han, Regular action in a ring with a finite number of orbits, Comm. Algebra 25 (1997), no. 7, 2227-2236.

8.
J. Han, Group actions in a unit-regular ring, Comm. Algebra 27 (1999), no. 7, 3353-3361.

9.
S. P. Redmond, The zero-divisor graph of non-commutative ring, Internat. J. Commutative Rings 1 (2002), no. 4, 203-211.

10.
S. P. Redmond, Structure in the zero-divisor graph of a noncommutative ring, Houston J. Math. 30 (2004), no. 2, 345-355.

11.
T. Wu, On directed zero-divisor graphs of finite rings, Discrete Math. 296 (2005), no. 1, 73-86.