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

Abstract

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).

Automorphisms of the Zero-Divisor Graph Over 2 × 2 Matrices

“…When reading the proof of [15, Theorem 3.8], we find some major mistakes there (the automorphism σ they constructed in the proof fails to be a bijection), which inspires us to determine the full automorphisms of Γ(M 2 (F q )) again. As an application of our main theorem, we show that [12,Theorem 3.9] and [15,Theorem 3.8] are both wrong.…”

“…[15, Theorem 3.8] (resp., [12,Theorem 3.9]) said that the automorphism group of Γ(R) (resp., Γ(M 2 (Z q )) where q is a prime) is isomorphic to the symmetric group S q+1 of degree q + 1, which means |Aut(Γ(R))| = (q + 1)! (resp., Aut(Γ(M 2 (Z q ))) = (q + 1)!).…”

“…In 2002, F. DeMeyer and K. Schneider [11] studied the relationship between Aut(R) and Aut(Γ(R)) when R is a commutative ring. In 2008, J. Han (see [12,Theorem 3.9]) showed that Aut(Γ(M 2 (Z p ))) is isomorphic to the symmetric group S p+1 of degree p + 1 when p is a prime. In 2011, S. Park and J. Han [15] generalized [12,Theorem 3.9] to an arbitrary finite field F q with q elements, and proved that Aut(Γ(M 2 (F q ))) ∼ = S q+1 .…”

“…In 2008, J. Han (see [12,Theorem 3.9]) showed that Aut(Γ(M 2 (Z p ))) is isomorphic to the symmetric group S p+1 of degree p + 1 when p is a prime. In 2011, S. Park and J. Han [15] generalized [12,Theorem 3.9] to an arbitrary finite field F q with q elements, and proved that Aut(Γ(M 2 (F q ))) ∼ = S q+1 . When reading the proof of [15, Theorem 3.8], we find some major mistakes there (the automorphism σ they constructed in the proof fails to be a bijection), which inspires us to determine the full automorphisms of Γ(M 2 (F q )) again.…”

The Connected Subgraph of the Torsion Graph of a Module

“…This concept also has been introduced and studied for semigroups by DeMeyer, McKenzie and Schneider in [13], and for near-rings by Cannon et al, in [12]. For recent developments on graphs of commutative rings see [3,4,5,17].…”