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 $\sum_ia_ix_i=0$.)