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An independence theorem and its consequences. (Russian) Zbl 0598.15002

It is well known that if \(A\) is a nilpotent operator of a vector space and \(A^ nv\neq 0\) then the vectors \(v, Av,\dots, A^ nv\) are linearly independent. Here the author proves the following generalization of this fact for the case of several operators:
Let \(A_{k_ 1}A_{k_ 2}\dots A_{k_ n}v\neq 0\) and the word \(B=A_{k_ 1}A_{k_ 2}...A_{k_ n}\) \((n=\| B\|)\) be a maximal one in the right lexicographic order among all words \(C\) which satisfy \(Cv\neq 0\) and \(\| C\| \leq n\). If all operators corresponding to subwords of the word \(B\) are nilpotent then the vectors \[ v,A_{k_ n}v,A_{k_{n-1}}A_{k_ n}v,\dots ,A_{k_ 1}A_{k_ 2}\dots A_{k_ n}v \] are linearly independent.
The theorem is valid for every field and for some rings. As an immediate consequence the author obtains two important theorems for the nilpotency of some finitely generated subalgebras of \(M_ n(K)\).
Reviewer: S.V.Mihovski

MSC:

16N40 Nil and nilpotent radicals, sets, ideals, associative rings
16S50 Endomorphism rings; matrix rings
15A03 Vector spaces, linear dependence, rank, lineability
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