Tarski-type problems for free associative algebras

In this paper we study fundamental model-theoretic questions for free associative algebras, namely, first-order classification, decidability of the first-order theory, and definability of the set of free bases. We show that two free associative algebras of finite rank over fields, at least one of which is infinite, are elementarily equivalent if and only if either their ranks are finite and equal or both are infinite and the fields are equivalent in the weak second order logic. In particular, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. We prove that if an arbitrary ring B with at least one Noetherian proper centralizer is first-order equivalent to a free associative algebra of finite rank over an infinite field then B is also a free associative algebra of finite rank over a field. This solves the elementary classification problem for free associative algebras in a wide class of rings (in fact, we note that a finitely generated algebra for which this result would not solve the classification problem would be a true monster). We also present a formula of the pure ring language which defines the set of free bases in a free associative algebra AK(X) with finite basis X over an infinite field K. To prove this result we establish rather deep connections between first order theory of AK(X) and the weak second order theory of the field K. In fact, we prove that AK(X) and the weak second order theory of K (the superstructure HF(K)) are bi-interpretable in each other. Finally, we show that the arithmetical hierarchy over AK(X) is proper, in particular there is no any reasonable “quantifier elimination” in free associative algebras.