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

Belov, A. Ya.

Mat. Sb., N. Ser. 135(177), No. 3, 373-384 (1988).

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

Keywords:

Shirshov bases of algebras; polynomial identities; complexity; bounded height; finitely generated alternative and Jordan PI-algebra