Commutative Subalgebras of Quantum Algebras

In the present paper, a general assertion is proved, claiming that, for every associative algebra $\mathscr A$ without zero divisors which admits a valuation and a seminorm concordant with the valuation, the transcendence degree of an arbitrary commutative subalgebra does not exceed the maximal number of independent pairwise pseudocommuting elements of some basis of the algebra $\mathscr A$. The author shows that for such a algebra $\mathscr A$ one can take an arbitrary algebra of quantum Laurent polynomials, quantum analogs of the Weyl algebra, and also some universal coacting algebras. In the case of the algebra $\mathscr L$ of quantum Laurent polynomials, it is proved that the transcendence degree of a maximal commutative subalgebra of $\mathscr L$ coincides with the maximal number of independent pairwise commuting elements of the monomial basis of the algebra $\mathscr L$.