the image $\pi(M)$ is entire over $\pi(K[X])$ is finite-dimensional over $\pi(K[X])$. The following example motivates this definition. Suppose $A={\Bbb Q}[x,1/x]$. Any projection $\pi$ such that $\pi(x)$ is algebraic has a finite-dimensional image. However the set $\{x\}$ is not an $s$-basis for ${\Bbb Q}[x,1/x]$. Thus boundedness of essential height is a non-commutative generalization of the property of [*entireness*]{}.

Shirshov bases consisting of words
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The Shirshov bases consisting of words are described by the following

\[ThBelheight\] A set $Y$ of words is a Shirshov basis for an algebra $A$ iff for any word $u$ of length not exceeding $m =
\Pid(A)$, the complexity of $A$, the set $Y$ contains a word cyclically conjugate to some power of $u$.

A similar result was obtained independently by G. P. Chekanu and V. Drensky. Problems related to local finiteness of algebras and to algebraic sets of words of degree not exceeding the complexity of the algebra were investigated in [@Ufn90; @Cio97; @Cio88; @CK93; @Che94; @Ufn85; @UC85]. Questions related to generalization of the independence theorem were considered in these papers as well.

Essential height
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Clearly the Gelfand — Kirillov dimension is estimated by the essential height. Furthermore an $s$-basis is a Shirshov basis iff 