Automorphisms of a certain subalgebra of the upper triangular matrix algebra

Authors : Özkay ÖZKAN, Mustafa AKKURT

Pages : 1150-1158

Doi:10.15672/hujms.513851

View : 26 | Download : 10

Publication Date : 2020-06-02

Article Type : Research Paper

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}\leftinsert ignore into journalissuearticles values( R\right); $, where $n$ is a positive integer greater than $1$. For the identity matrix $I$, $\alpha \in R$, $A \in N_ninsert ignore into journalissuearticles values(R);$, the set of all elements $\alpha I+A$ is defined as the scalar upper triangular matrix algebra $ST_ninsert ignore into journalissuearticles values(R);$ which is a subalgebra of the upper triangular matrices $T_ninsert ignore into journalissuearticles values(R); .$ In this paper, we investigate the $R$-algebra automorphisms of $ST_{n}\leftinsert ignore into journalissuearticles values( R\right); .$ We extend the automorphisms of $N_{n}\leftinsert ignore into journalissuearticles values( R\right); $ to $ST_{n}\leftinsert ignore into journalissuearticles values( R\right);$ and classify all the automorphisms of $ST_{n}\leftinsert ignore into journalissuearticles values( R\right); .$ We generalize the results of Cao and Wang and prove that not all automorphisms of $ST_{n}\leftinsert ignore into journalissuearticles values( R\right); $ can be extended to the automorphisms of $T_{n}insert ignore into journalissuearticles values(R);.$
Keywords : triangular matrix algebra, upper triangular matrix algebra, automorphisms