Elementary Equivalence of Categories of Modules over Rings, Endomorphism   Rings, and Automorphism Groups of Modules
  In this paper we give a small review of some recent results of elementary equivalence of linear and algebraic groups and our last new results of elementary equivalence of categories of modules, endomorphism rings of modules, lattices of submodules of modules, and automorphism groups of modules. 
E.I.Bunina, A.V.Mikhalev