The group of automorphisms of a zero-divisor graph based on rank one upper triangular matrices

Maps on upper triangular matrices preserving zero products

“…Using the main theorem of the latter paper and some other arguments, Wang in [14] described all the automorphisms of the zero divisor graphs of T n (F ). The results from [16] and [14] are valid when F is a finite field. In this article we will show how we can easily extend them to the case when F is infinite and, thanks to it, obtain the description of all the maps preserving zero products on T n (F ).…”

“…One can see that if we know the form of all automorphisms of a zero-divisor graph of some ring R, then we know the form all the bijective maps that preserve zero products on R. Therefore such automorphisms are of our interest. In [16] we can find how the automorphisms of the zero divisor graph of T n (F ) act on rank one triangular matrices. Using the main theorem of the latter paper and some other arguments, Wang in [14] described all the automorphisms of the zero divisor graphs of T n (F ).…”

“…As we have already mentioned, the proof of our theorem will be based on the results of Wang in [14] and Wong, Ma and Zhou in [16]. First we cite the theorem from the former.…”

“…The field F is finite. The proof is based on the result from [16] in which this assumption appears and on some results of the author where this assumption is not necessary. Hence, we can make use of Proposition 2.1 on the condition that the result from [16] will hold for any field.…”

Automorphism Group of the Subspace Inclusion Graph of a Vector Space

“…Park and Han [8] proved that Aut(Γ(R)) ∼ = S q+1 for R = M 2 (F q ) with F q an arbitrary finite field. In [11], Wong et al determined the automorphisms of the zerodivisor graph with vertex set of all rank one upper triangular matrices over a finite field. By applied the main theorem in [11], Wang [9] and [10] respectively determined the automorphisms of the zero-divisor graph defined on all n × n upper triangular matrices or on all n × n full matrices when the base field is finite.…”