On a Shirshov basis of relatively free algebras of complexity $n$

A Shirshov basis is a set of elements of an algebra $A$ over which $A$ 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 $\Phi$ with unity. It is proved that the set of monomials of degree at most $m^2$ is a Shirshov basis in a Jordan PI-algebra of degree $m$. It is shown that under certain conditions on $\operatorname{var}(B)$ (satisfied by alternative and Jordan PI-algebras), if each factor of $B$ with nilpotent projections of all elements of $M$ is nilpotent, then $M$ is a Shirshov basis of $B$ if $M$ generates $B$ as an algebra.
Bibliography: 12 titles.