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On a Shirshov base with respect to free algebras of complexity n. (Russian) Zbl 0659.16012

The author proves several significant results on Shirshov bases of algebras satisfying polynomial identities. It is shown that a finitely generated associative PI-algebra A of complexity n has bounded height over the set of monomials of length \(\leq n\). It follows that if A is of degree m, then A has bounded height over monomials of length \(\leq [m/2]\). A finitely generated alternative and Jordan PI-algebra B of degree m is shown to be of bounded height over the set of monomials of length \(\leq m^ 2\). Shirshov’s bases consisting of words are described for relatively free associative, and alternative, finitely generated algebras.
Reviewer: J.Okniński

MSC:

16Rxx Rings with polynomial identity
17C10 Structure theory for Jordan algebras
17D05 Alternative rings
17C05 Identities and free Jordan structures
17A50 Free nonassociative algebras
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