Van den Dries, L. and J. Holly, Quantifier elimination for modules with scalar variables, Annals of Pure and Applied Logic 57 (1992) 161-179. We consider modules as two-sorted structures with scalar variables ranging over the ring. We show that each formula in which all scalar variables are free is equivalent to a formula of a very simple form, uniformly and effectively for all torsion-free modules over gcd domains (=Bezout domains expanded by gcd operations). For the case of Presburger arithmetic with scalar variables the result takes a still simpler form, and we derive in this way the polynomial-time decidability of the sets defined by such formulas.
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