In the Wikipedia article (<https://en.wikipedia.org/wiki/Quantifier_elimination#cite_note-4>) it is said that every abelian group has quantifier elimination property and a long old paper of W. Szmielew (1955) is given as a reference. But as far as I know, this is true for special classes of abelian groups (like divisible, ordered, ... groups). I am not an expert of model theory but I need to apply quantifier elimination for reduced abelian groups (abelian groups the only divisible subgroup in which is the trivial one), so I need to know if really we have quantifier elimination for any abelain group and if not, what is the most complex quantifier combination of formulas in such groups.



Quantifier elimination for abelian groups
Abelian groups are the same thing as $\mathbb Z$-modules. In general, for any ring $R$, the theory of left $R$-modules has quantifier elimination down to Boolean combinations of primitive positive formulas and certain sentences (expressing so-called Baur–Monk invariants). This is the Baur–Monk quantifier elimination theorem; see e.g. §A.1 in Hodges, *Model Theory*.


In particular, since the theory of any particular abelian group or $R$-module is complete, it has quantifier elimination down to Boolean combinations of p.p. formulas.


Note that a p.p. formula expresses solvability of a linear system; in the case where $R$ is a PID (including abelian groups with $R=\mathbb Z$), the Smith normal form shows that you can reduce to the case of p.p. formulas with one quantifier. That is, every formula $\phi(\vec x)$ is equivalent to a Boolean combination of invariant sentences and the divisibility formulas $a\mid\sum\_ia\_ix\_i$ for some $a,a\_i\in R$. (For $a=0$, this amounts to the original atomic formulas $\sum\_ia\_ix\_i=0$.)


