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 $T_{n}(R).$
📌 3 Maddede Bu Makale
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Amaç & Hipotez: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 $1$.
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Temel Bulgular & Veri: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 $1$.
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Sonuç & Katkı: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 $T_{n}(R).$
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 $T_{n}(R).$
ÖZKAN, Ö., AKKURT, M. (2022). Automorphisms of a certain subalgebra of the upper triangular matrix algebra. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 3, 1150–1158. https://ajindex.com/articles/529600
Y. Cao and J. Wang, A note on algebra automorphisms of strictly upper triangular matrices over commutative rings, Linear Algebra Appl. 311, 187–193, 2000
S.P. Coelho, Automorphism group of certain algebras of triangular matrices, Arch. Math. 61, 119–123, 1993
T.P. Kezlan, A note on algebra automorphisms of triangular matrices over commutative rings, Linear Algebra Appl. 135, 181–184, 1990.