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Preprint Number 923

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923. Olga Kharlampovich and Alexei Myasnikov
Tarski-type problems for free associative algebras

Submission date: 14 September 2015.


We consider some fundamental model-theoretic questions that can be asked about a given algebraic structure (a group, a ring, etc.), or a class of structures, to understand its principal algebraic and logical properties. These Tarski type questions include: elementary classification and decidability of the first-order theory.
We describe solutions to Tarski's problems in the class of free associative algebras, and some open problems. We will show that unlike free groups, two free associative algebras of finite rank over the same field are elementarily equivalent if and only if they are isomorphic. Two free associative algebras of finite rank over different infinite fields are elementarily equivalent if and only if the fields are equivalent in the weak second order logic, and the ranks are the same. We will show that if a ring B, which has 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 over a field with a finite basis. We will also show that for any field the theory of a free associative algebra is undecidable.

Mathematics Subject Classification: 16B70

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Full text arXiv 1509.04112: pdf, ps.

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