height theorem for Lie algebras with a sparse identity. In the paper by one of the authors [@Belov1] the height theorem was proved for some class of rings asymptotically close to associative rings. In particular, this class contains alternative and Jordan $\PI$-algebras.

Shirshov bases
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Suppose $A$ is a $\PI$-algebra and a subset $M\subseteq A$ is its $s$-basis. Then if all elements of $M$ are algebraic over $K$ then $A$ is finite-dimensional (the Kurosh problem). Boundedness of essential height over $Y$ implies “the positive solution of the Kurosh problem over $Y$”. The converse is less trivial.

\[A. Ya. Belov\] \[ThKurHmg\] a) Suppose $A$ is a graded $\PI$-algebra, $Y$ is a finite set of homogeneous elements. Then if for all $n$ the algebra $A/Y^{(n)}$ is nilpotent then $Y$ is an $s$-basis for $A$. If moreover $Y$ generates $A$ as an algebra then $Y$ is a Shirshov basis for $A$.

b\) Suppose $A$ is a $\PI$-algebra, $M\subseteq A$ is a Kurosh subset in $A$. Then $M$ is an $s$-basis for $A$.

Let $Y^{(n)}$ denote the ideal generated by $n$th powers of elements from $Y$. A set $M\subset A$ is called [*a Kurosh set*]{} if any projection $\pi\colon A\otimes K[X]\to A'$ such that 