the set of $i$ such that $t_i\not\in D$ contains not more than $h$ elements. The [*essential height*]{} of a set of words is defined similarly.

Informally speaking, any long word is a product of periodic parts and ”gaskets” of restricted length. The essential height is the number of periodic parts, and the ordinary height accounts “gaskets” as well.

The height theorem suggests the following questions:

1.  To which classes of rings the height theorem can be extended?

2.  Over which $Y$ the algebra $A$ has bounded height? In particular, what sets of words can be taken for $\{v_i\}$?

3.  What is the structure of the degree vector $(k_1,\ldots,k_h)$? First of all, what sets of its components are essential, that is, what sets of $k_i$ can be unbounded simultaneously? What is the value of essential height? Is it true that the set of degree vectors has some regularity properties?

4.  What estimates for the height are possible?

Let us discuss the above questions.

Non-associative generalizations
-------------------------------

The height theorem was extended to some classes of near-associative rings. S. V Pchelintsev [@Pchelintcev] has proved it for the alternative and the $(-1,1)$ сases, S. P. Mishchenko [@Mishenko1] has obtained an analogue of the 