Regular action in a ring with a finite number of orbits

Abstract. Let R be a ring with identity, X the set of all nonzero, nonunits of R and G the group of all units of R. First, we investigate some connected conditions of the zero-divisor graph Γ(R) of a noncommutative ring R as follows: (1) if Γ(R) has no sources and no sinks, then Γ(R) is connected and diameter of Γ(R), denoted by diam(Γ(R)) (resp. girth of Γ(R), denoted by g(Γ(R))) is equal to or less than 3; (2) if X is a union of finite number of orbits under the left (resp. right) regular action on X by G, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3, in addition, if R is local, then there is a vertex of Γ(R) which is adjacent to every other vertices in Γ(R); (3) if R is unitregular, then Γ(R) is connected and diam(Γ(R)) (resp. g(Γ(R))) is equal to or less than 3. Next, we investigate the graph automorphisms group of Γ(Mat 2 (Zp)) where Mat 2 (Zp) is the ring of 2 by 2 matrices over the galois field Zp (p is any prime).

“…Let X be a union of n orbits under the left (resp. right) regular action on X by G. Since R is a local ring, by [8,Lemma 2.3] there exists x ∈ X such that…”

Section: Theorem 210 Let R Be a Ring Such That X Is A Union Of Finimentioning

confidence: 99%

Section: 2] If γ(R) Contains a Cycle Then 1 + 2diam(γ(r)) ≥ G(γ(r))mentioning

confidence: 99%

The Zero-Divisor Graph Under Group Actions in a Noncommutative Ring

Han¹

2008

“…In [2], the rings with a finitely many orbits under the left regular action on X by G are considered and some properties are obtained. By Hirano [3], a full characterization of these rings was given, in that they are precisely rings that are a product of a finite ring and finitely many left Artinian left uniserial rings.…”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

Rings With a Finite Number of Orbits Under the Regular Action

Han¹,

Park²

2014

Self Cite

Abstract. Let R be a ring with identity, X(R) the set of all nonzero, non-units of R and G(R) the group of all units of R. We show that for a matrix ring Mn(D), n ≥ 2, if a, b are singular matrices of the same rank, then |o ℓ (a)| = |o ℓ (b)|, where o ℓ (a) and o ℓ (b) are the orbits of a and b, respectively, under the left regular action. We also show that for a semisimple Artinian ring R such that, with D i infinite division rings of the same cardinalities or R is isomorphic to the ring of 2 × 2 matrices over a finite field if and only if |o ℓ (x)| = |o ℓ (y)| for all x, y ∈ X(R).

“…Recall that G is transitive on X (or G acts transitively on X) if there is an x ∈ X with o(x) = X and the group action on X by G is trivial if o(x) = {x} for all x ∈ X. In [7], it has been shown that if X is a union of a finite n number of orbits under the regular action of G on X, then (1) x n+1 = 0 for all x ∈ J, and X is the set of all nonzero left zero-divisors of R; (2) R is a local ring, J n = (0) and J n+1 = (0) if and only if there exists…”

Section: Introduction and Basic Definitionsmentioning

confidence: 99%

The Zero-Divisor Graph Under a Group Action in a Commutative Ring

Han¹

2010

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 Γ(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 Γ(R) which is adjacent to every other vertex in Γ(R) if and only if R is a local ring or R Z 2 × 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, J 2 , . . . , 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.