Let G be a finite group acting linearly on the vector space V over a field of arbitrary characteristic. The action is called coregular if the invariant ring is generated by algebraically independent homogeneous invariants and the direct summand property holds if there is a surjective k[V]^G-linear map π:k[V]→ k[V]^G. The following Chevalley--Shephard--Todd type theorem is proved. Suppose G is abelian, then the action is coregular if and only if G is generated by pseudo-reflections and the direct summand property holds.