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Modular automorphisms preserving idempotence and Jordan isomorphisms of triangular matrices over commutative rings. (English) Zbl 1003.15001

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

MSC:

15A04 Linear transformations, semilinear transformations
16S50 Endomorphism rings; matrix rings
15B33 Matrices over special rings (quaternions, finite fields, etc.)
16W20 Automorphisms and endomorphisms
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