Automorphisms of a certain subalgebra of the upper triangular matrix algebra

Abstract

For a commutative ring $R$ with unity, the $R$-algebra of strictly upper triangular $n\times n$ matrices over $R$ is denoted by $N_{n}\left( R\right) $, where $n$ is a positive integer greater than $1$. For the identity matrix $I$, $\alpha \in R$, $A \in N_n(R)$, the set of all elements $\alpha I+A$ is defined as the scalar upper triangular matrix algebra $ST_n(R)$ which is a subalgebra of the upper triangular matrices $T_n(R) .$ In this paper, we investigate the $R$-algebra automorphisms of $ST_{n}\left( R\right) .$ We extend the automorphisms of $N_{n}\left( R\right) $ to $ST_{n}\left( R\right)$ and classify all the automorphisms of $ST_{n}\left( R\right) .$ We generalize the results of Cao and Wang and prove that not all automorphisms of $ST_{n}\left( R\right) $ can be extended to the automorphisms of $T_{n}(R).$