On a shirshov basis of relatively free algebras of complexity n

A Shirshov basis is a set of elements of an algebra over which has bounded height in the sense of Shirshov. A description is given of Shirshov bases consisting of words for associative or alternative relatively free algebras over an arbitrary commutative associative ring with unity. It is proved that the set of monomials of degree at most is a Shirshov basis in a Jordan PI-algebra of degree. It is shown that under certain conditions on (satisfied by alternative and Jordan PI-algebras), if each factor of with nilpotent projections of all elements of is nilpotent, then is a Shirshov basis of if generates as an algebra.