Automorphisms of a certain subalgebra of the upper triangular matrix algebra

Year 2020, Volume: 49 Issue: 3, 1150 - 1158, 02.06.2020

Özkay Özkan , Mustafa Akkurt

https://doi.org/10.15672/hujms.513851

https://izlik.org/JA35MY99RD

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 $T_{n}(R).$

Keywords

triangular matrix algebra , upper triangular matrix algebra , automorphisms

References

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