## 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
Full Text: