Clinical evidence has shown that rib-suppressed chest X-rays (CXRs) can improve the reliability of pulmonary disease diagnosis. However, previous approaches on generating rib-suppressed CXR face challenges in preserving details and eliminating rib residues. We hereby propose a GAN-based disentanglement learning framework called Rib Suppression GAN, or RSGAN, to perform rib suppression by utilizing the anatomical knowledge embedded in unpaired computed tomography (CT) images. In this approach, we employ a residual map to characterize the intensity difference between CXR and the corresponding rib-suppressed result. To predict the residual map in CXR domain, we disentangle the image into structure- and contrast-specific features and transfer the rib structural priors from digitally reconstructed radiographs (DRRs) computed by CT. Furthermore, we employ additional adaptive loss to suppress rib residue and preserve more details. We conduct extensive experiments based on 1,673 CT volumes, and four benchmarking CXR datasets, totaling over 120K images, to demonstrate that (i) our proposed RSGAN achieves superior image quality compared to the state-of-the-art rib suppression methods; (ii) combining CXR with our rib-suppressed result leads to better performance in lung disease classification and tuberculosis area detection.