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Modular automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings. (English) Zbl 1003.15001
Authors’ abstract: Let ${\cal R}$ be a commutative ring with 1 and 2 being the units of ${\cal R}$, let $T_n({\cal R})$ be the $n\times n$ upper triangular matrix module over ${\cal R}$, and let ${\cal L}({\cal R})$ be the set of all ${\cal R}$-module automorphisms on $T_n({\cal R})$, which preserve idempotence. The main result of this paper is: if $f$ is an ${\cal R}$-module automorphism on $T_n({\cal R})$, then $f\in{\cal L}({\cal R})$ if and only if there exist an invertible matrix $U\in T_n({\cal R})$ and an idempotent element $e\in{\cal R}$ such that $f(X)= U(eX+(1-e) X^\delta)U^{-1}$ for any $X= (x_{ij}) \in T_n({\cal R})$, where $X^\delta= (x_{n+1-j,n+1-i})$. As applications, we determine all Jordan isomorphisms of $T_n({\cal R})$ over ${\cal R}$.

15A04Linear transformations, semilinear transformations (linear algebra)
16S50Endomorphism rings: matrix rings
15B33Matrices over special rings (quaternions, finite fields, etc.)
16W20Automorphisms and endomorphisms of associative rings
Full Text: DOI
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