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On local finiteness of algebras. (Russian) Zbl 0681.16010
The author establishes some sharpening of existing results of Burnside type. In the following F denotes a field. Theorem 1: Let \(B_ 1,...,B_ m\in M_ n(F)\), \(A\in \{B_ 1,...,B_ m\}\), and let k be the number of nonzero Jordan blocks of A. If all the products \(\prod \{A^{p_ i}B_ 1^{q_{i1}}...B_ m^{q_{im}}:\) \(1\leq i\leq k\}\) are nilpotent, where the sum of \(p_ i\) is at most the rank of A and the sum of \(q_{ij}\) is at most n-rank A, then \(\{B_ 1,...,B_ m\}\) generates a nilpotent subalgebra of \(M_ n(F)\). Theorem 2: Let R be a PI-algebra over F of complexity n. If all products of elements of R of length at most n are algebraic over F, then R is locally finite-dimensional.
Reviewer: A.A.Iskander

MSC:
16Rxx Rings with polynomial identity
16S50 Endomorphism rings; matrix rings
15A23 Factorization of matrices
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