Geodesic
Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules
Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 2, pp. 51-134

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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. 
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     author = {E. I. Bunina and A. V. Mikhalev},
     title = {Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules},
     journal = {Fundamentalʹna\^a i prikladna\^a matematika},
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     year = {2004},
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E. I. Bunina; A. V. Mikhalev. Elementary equivalence of categories of modules over rings, endomorphism rings, and automorphism groups of modules. Fundamentalʹnaâ i prikladnaâ matematika, Tome 10 (2004) no. 2, pp. 51-134. http://geodesic.mathdoc.fr/item/FPM_2004_10_2_a2/