Given a map \(f:X\rightarrow Y\) of locally compact Hausdorff spaces, \textit{W. S. Massey} [Algebraic topology: an introduction (1967; Zbl 0153.24901)] has shown that if \(f\) induces an isomorphism \(H^n(X;{\mathbb{Z}})\cong H^n(Y;{\mathbb{Z}})\) for all \(n\geq 0\), then \(f\) induces an isomorphism in homology with coefficients in any abelian group. The authors generalize this result to show that for any non-complete principal ideal domain \(R\), a map \(f:X\rightarrow Y\) of chain complexes of \(R\)-modules that yields an isomorphism in cohomology with coefficients in \(R\) also produces an isomorphism in homology with coefficients in any \(R\)-module. This result is obtained by proving that if \(\Hom_R(M,R)=0\) and \(\text{Ext}_R(M,R)=0\), then \(M=0\) for any \(R\)-module \(M\). The authors use the main result to obtain a version of the Dual Whitehead Theorem due to \textit{H. J. Baues} [Obstruction theory on homotopy classification of maps, Lect. Notes Math. 628 (1977; Zbl 0361.55017)]. In particular, they establish that a map \(f:X\rightarrow Y\) of \(R\)-Postnikov spaces of order \(k\geq 1\) that induces an isomorphism in cohomology must be a weak homotopy equivalence.