Automorphisms of the co-maximal ideal graph over matrix ring

Abstract

Let [Formula: see text] be a finite field with [Formula: see text] elements, [Formula: see text] be the ring of all [Formula: see text] matrices over [Formula: see text], [Formula: see text] be the set of all nontrivial left ideals of [Formula: see text]. The co-maximal ideal graph of [Formula: see text], denoted by [Formula: see text], is a graph with [Formula: see text] as vertex set and two nontrivial left ideals [Formula: see text] of [Formula: see text] are adjacent if and only if [Formula: see text]. If [Formula: see text], it is easy to see that [Formula: see text] is a complete graph, thus any permutation of vertices of [Formula: see text] is an automorphism of [Formula: see text]. A natural problem is: How about the automorphisms of [Formula: see text] when [Formula: see text]. In this paper, we aim to solve this problem. When [Formula: see text], a mapping [Formula: see text] on [Formula: see text] is proved to be an automorphism of [Formula: see text] if and only if there is an invertible matrix [Formula: see text] and a field automorphism [Formula: see text] of [Formula: see text] such that [Formula: see text] for any [Formula: see text], where [Formula: see text] and [Formula: see text] for [Formula: see text].