The Tarski Theorems and Elementary Equivalence of Group Rings

The Tarski theorems, proved by Myasnikov and Kharlampovich and inde-pendently by Sela say that all nonabelian free groups satisfy the same first-order or elementary theory. Kharlampovich and Myasnikov also prove that the elementary theory of free groups is decidable. For a group ring they have proved that the first-order theory (in the language of ring theory) is not decidable and have studied equations over group rings, especially for torsion-free hyperbolic groups. In this note we examine and survey extensions of Tarksi-like results to the collection of group rings and examine relationships between the universal and elementary theories of the corresponding groups and rings and the corresponding universal theory of the formed group ring. To accomplish this we introduce different first-order languages with equality whose model classes are respectively groups, rings and group rings. We prove that if R[G] is elementarily equivalent to S[H] then simultaneously the group G is elementarily equivalent to the group H and the ring R is elementarily equivalent to the ring S with respect to the appropriate languages. Further if G is universally equivalent to a nonabelian free group F and R is universally equivalent to the integers Z then R[G] is universally equivalent to Z[F] again with respect to an ap-propriate language.

[1]  B. H. Neumann,et al.  On Ordered Groups , 1949 .

[2]  Benjamin Baumslag.  Residually Free Groups , 1967 .

[3]  J. Bell,et al.  Models and ultraproducts , 1971 .

[4]  Saharon Shelah,et al.  Every two elementarily equivalent models have isomorphic ultrapowers , 1971 .

[5]  D. Passman,et al.  The algebraic structure of group rings , 1977 .

[6]  ∃-Free groups , 1989 .

[7]  Anthony Gaglione,et al.  Even More Model Theory of Free Groups , 1993 .

[8]  Alexei G. Myasnikov,et al.  Irreducible Affine Varieties over a Free Group: I. Irreducibility of Quadratic Equations and Nullstellensatz , 1998 .

[9]  Alexei G. Myasnikov,et al.  Irreducible Affine Varieties over a Free Group: II. Systems in Triangular Quasi-quadratic Form and Description of Residually Free Groups , 1998 .

[10]  Z. Sela,et al.  Diophantine geometry over groups I: Makanin-Razborov diagrams , 2001 .

[11]  A. Myasnikov,et al.  Implicit function theorem over free groups , 2003, math/0312509.

[12]  Z. Sela.  Diophanting geometry over groups II: Completions, closures and formal solutions , 2003 .

[13]  A. Myasnikov,et al.  Groups St Andrews 2001 in Oxford: Elementary theory of groups , 2003 .

[14]  A. Myasnikov,et al.  Effective JSJ Decompositions , 2004, math/0407089.

[15]  Z. Sela,et al.  Diophantine geometry over groups IV: An iterative procedure for validation of a sentence , 2004 .

[16]  Z. Sela.  Diophantine geometry over groups III: Rigid and solid solutions , 2005 .

[17]  Z. Zela.  Diophantine geometry over groups V1: Quantifier elimination I , 2005 .

[18]  A. Myasnikov,et al.  Elementary theory of free non-abelian groups , 2006 .

[19]  A. Myasnikov,et al.  The Elementary Theory of Groups: A Guide Through the Proofs of the Tarski Conjectures , 2014 .

[20]  The Tarski Problems and Their Solutions , 2015 .