Automorphisms of Zero Divisor Graphs of Square Radical Zero Commutative Unital Finite Rings

There has been extensive research on the structure of zero divisors and units of commutative ﬁnite rings. However, the classiﬁcation of such rings via a well known structure of zero divisors has not been done in general. More speciﬁcally, the automorphisms of such classes of rings have not been fully characterized. In this paper we obtain a more complete illustration of the automorphisms of zero divisor graphs of ﬁnite rings in which the product of any zero divisor is zero.

Introduction The classification of completely primary finite rings has received much attention in the recent years.
For related studies see [1,2,3,4,5,6]. Most scholars have concentrated in obtaining the structures of zero divisor graphs of commutative finite rings. However, little research has been done on the automorphisms of zero divisor graphs of completely primary finite rings ,despite the fact that automorphisms have played practical role in understanding complexity of many algebraic structures. In view of the above, this research investigated the automorphsms of zero divisor graphs of square radical zero commutative unital finite rings. In this study, R shall denote commutative finite rings as constructed in the sequel, Z(R) will represent the Jacobson radical of R, the automorphism of the zero divisor graph of the ring R will be denoted by Aut(Γ(R)). The edge between vertices of a graph will be denoted E and |E| will represent the cardinality of the graph while v denote a vertex in a graph. The cardinality of the automorphism group shall be denoted by |Aut(Γ(R))|. Unless otherwise given, GR(p nr , p n ) will denote Galois ring of order p nr and characteristic p n and it is uniquely determined by the invariant p, n and r. The following result due to Raghavendran will be useful in the sequel.

Preliminary Results
Theorem 1.1. ( [5] pp 99) Let R be a finite ring with multiplicative identity 1 = 0 whose zero divisors form an additive group Z(R). Then, (i) Z(R) is the Jacobson radical of R (ii) |R| = p nr , and |Z(R)| = p (n−1)r for some prime integer p and some positive integers n and r (iii) (Z(R)) n = (0) (iv) The characteristic of a ring R if p k for some positive integer k with 1 ≤ k ≤ n; and (v) If the characteristic is p n , then R is commutative.
We begin with construction of the ring which can be obtained in [5].

The Construction of Square Radical Zero Finite Rings
For every prime integer p and positive integer r, let R 0 = GF (p kr , p k ) where k = 1, 2. For all i = 1, . . . , h, let u i ∈ Z(R) and U is an h-dimensional R 0 -module generated by {u 1 , . . . , u h }. Suppose R = R 0 ⊕ U is an additive group, it can easily be shown that the multiplication 3 Characteristic of R is p Proposition 1. Let R be a ring of characteristic p in the Construction. Then, Proof. In the above construction we consider k = 1 so that R 0 = GR(p r , p). Let K = R 0 /pR 0 be the field. Suppose U = K h is an R 0 -module generated by {u 1 , u 2 , u 3 , · · · , u h }. Therefore R = R 0 ⊕ U is an additive abelian group. But Z(R) = R 0 u 1 ⊕ R 0 u 2 ⊕ · · · ⊕ R 0 u h and every element in Z(R) * is of the form (0, a 1 , a 2 , · · · , a h ) so that the product of every pair (0, a 1 , a 2 , · · · , a h ), Dividing |Aut(Γ(R))| by Φ(p hr ) and expressing the relation in terms of |Aut(Γ(R))| establishes the relation |Aut(Γ(R))| = Proof. This is a consequence of Proposition 1.

Characteristics of R is p 2
Proposition 7. Let R be a ring in the Construction of characteristic p 2 . Then, Proof. In the above Construction , we have R 0 = GR(p 2r , p 2 ) and K = R 0 /pR 0 . Let U = K h be an R 0 -module generated by {u 1 , u 2 , u 3 , · · · , u h } and R = R 0 ⊕U is an additive group. Clearly, Z(R) = pR 0 ⊕R 0 u 1 ⊕R 0 u 2 ⊕· · ·⊕R 0 u h and the product of every pair of elements in Z(R) is zero. Thus |R| = |R 0 ||U | = p (h+2)r ⇒ |Z(R)| = p (h+1)r and |V (Γ(R))| = |Z(R) * | = p (h+1)r − 1. Since every element of Z(R) * is adjacent to all the other vertices of the zero divisor graph of R, so Aut(Γ(R)) must permute all the symmetries of Γ(R) independently so that Aut(Γ(R)) ∼ = Proof. The result follows from Proposition 7.