Chain equivalence proof

Question

Let be $R$ a principal ideal domain. Let be $h: C \to D $ a homomorphism between complex chains, so $h= \{f_n: C_n \to D_n | n \in \mathbb{Z}\}$ and $h$ induces an isomorphism $g: H_n (C) \to H_n (D)$. Where $H_n(C)= Ker(\partial) / Im(\partial)$ and also $C_n$ and $D_n$ are free modules, so every submodule of $C_n$ and $D_n$ is free because $R$ a principal ideal domain. I have to prove that $h$ is a chain equivalence in other words I need to prove that exists $k: D \to C$ such that $k \circ h$ and $h \circ k$ are homotopic to the identity of $C$ and $D$