  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 are elementarily
equivalent if and only if their ranks are the same 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. Finally, we present a formula of the ring language which defines the
set of free bases in a free associative algebra of finite rank.
