Abstract
The first result in elementary equivalence of linear groups was proved by A.I.\,Maltsev in~1961. He proved that the group G(m,K)$ is elementarily equivalent to the group $G(n,L) (G=GL,PGL,SL,PSL, m,n>2, K and L are fields of characteristic~0} iff m=n and the fields K and L are elementarily equivalent.
In 1992 C.I.Beidar and A.V.Mikhalev using some results in model theory (namely, the construction of ultrapowers) formulated a general approach to problems of elementary equivalence of different algebraic structures. Taking into account some results in the theory of linear groups over rings, they obtained easy proofs of Maltsev-type theorems in rather general situations (for linear groups over prime rings, for multiplicative semigroups, lattices of submodules, and so on).
In 1998--2001 E.I. Bunina continued to study elementary properties of linear. In 1998 the results of A.I.Maltsev were generalized to unitary linear groups over fields with involution. As it was done for linear groups over rings, using the construction of ultraproducts, E.I.Bunina in 1998 considered elementary equivalence of unitary linear groups over rings and skewfields with involutions.
She proved two following theorem:
1. If K and K' are associative (commutative) rings with 1/2 and 1/3, j and j' are involutions in the rings K and K', respectively, and n,m>2 (n,m>1), then the unitary linear groups $U(2n,K,j,Q(2n)) and U(2m,K',j',Q(2m)) are elementarily equivalent iff the rings M(2n,K) and M(2m,K') are elementarily equivalent as rings with involutions (induced from K and K'), respectively.
2. If skewfields (fields) F and F' have characteristic which is not equal to 2, j and j' are involutions in skewfields (fields) F and F' respectively, and n,m>2 (n,m>1), then the unitary linear groups U(2n,F,j,Q(2n)) and U(2m,F',j',Q(2m)) are elementarily equivalent iff the skewfields (fields) F and F' are elementarily equivalent as the skewfields (fields) with involutions j and j' respectively.
In 2001-2004 E.I.Bunina studied elementary properties of Chevalley groups over fields. The class of all Chevalley groups contains many classical groups like SL(n,K), PSL(n,K), SO(n,K), Spin(n,K), PSO(n,K), Sp(2n,K), PSp(2n,K). Therefore the studied groups intersect with the groups which were considered by A.I.Maltsev, but there are many other algebraicgroups in this class.
The main result is the following theorem:
Suppose that universal (adjoint) Chevalley groups G and G' are constructed respectively by fields K and K' of characteristic not equal to 2, and simple Lie algebras L and L'. Then g and G' are elementarily equivalent iff K and K' are elementarily equivalent and the algebras L and L' are isomorphic.
In 2003-2004 E.I.Bunina and A.V.Mikhalev considered elementary properties of categories of modules over rings, endomorphism rings of almost free modules of infinite ranks over rings and automorphism groups of almost free modules of infinite ranks over rings. For example, they proved that two categories mod-R and mod-R' (R and R' are commutative, local, semilocal, Artinean rings) are elementarily equivalent iff the rings R and R' are equivalent in the second order logic. The same resultes were obtained for endomorphism rings and automorphisms groups of free modules of infinite ranks over different classes of rings.