Annals of Pure and Applied Logic 57 (2):161-179 (1992)

Van den Dries, L. and J. Holly, Quantifier elimination for modules with scalar variables, Annals of Pure and Applied Logic 57 161–179. 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 . 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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DOI 10.1016/0168-0072(92)90025-u
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Model Theory of Modules.Martin Ziegler - 1984 - Annals of Pure and Applied Logic 26 (2):149-213.
Model Theory and Modules.Mike Prest - 1989 - Journal of Symbolic Logic 54 (3):1115-1118.

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