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Let $R$ be a commutative ring with identity $1$, $n\geq 3$, and let $\mathcal{T}_n(R)$ be the linear Lie algebra of all upper triangular $n\times n$ matrices over $R$. A linear map $\varphi$ on $\mathcal{T}_n(R)$ is called to be strong commutativity preserving if $[\varphi(x),\varphi(y)]=[x,y]$ for any $x,y\in \mathcal{T}_n(R)$. We show that an invertible linear map $\varphi$ preserves strong commutativity on $\mathcal{T}_n(R)$ if and only if it is a composition of an idempotent scalar multiplication, an extremal inner automorphism and a linear map induced by a linear function on $\mathcal{T}_n(R)$.
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참고문헌 (14)
참고문헌 신청
H. E. Bell / 1994 / On commutativity and strong commutativity-preserving maps / Canad. Math. Bull 37(4):443~447
Dominik Benkovič / 2004 / Commuting traces and commutativity preserving maps on triangular algebras / Journal of Algebra 280(2):797~824
Matej Bresar / 1993 / Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings / Transactions of the American Mathematical Society 335(2):525~546
Matej Brešar / 1994 / Strong Commutativity Preserving Maps of Semiprime Rings / Canadian Mathematical Bulletin 37(4):457~460
Matej Brešar / 2005 / Commutativity preserving linear maps on central simple algebras / Journal of Algebra 284(1):102~110
Dominik Benkovič / 2004 / Commuting traces and commutativity preserving maps on triangular algebras / Journal of Algebra 280(2):797~824
Matej Bresar / 1993 / Commuting Traces of Biadditive Mappings, Commutativity-Preserving Mappings and Lie Mappings / Transactions of the American Mathematical Society 335(2):525~546
Matej Brešar / 1994 / Strong Commutativity Preserving Maps of Semiprime Rings / Canadian Mathematical Bulletin 37(4):457~460
Matej Brešar / 2005 / Commutativity preserving linear maps on central simple algebras / Journal of Algebra 284(1):102~110
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