  Let $f: X \rightarrow A$ be an abelian cover from a complex algebraic variety
with quotient singularities to an abelian variety. We show that $f^*$ induces
an isomorphism between the rational cohomology rings $H^\bullet(A, \mathbb{Q})$
and $H^\bullet(X, \mathbb{Q})$ if and only if $f^*$ induces an isomorphism
between the Chow rings with rational coefficients
$\mathrm{CH}^\bullet(A)_{\mathbb{Q}}$ and
$\mathrm{CH}^\bullet(X)_{\mathbb{Q}}$.
