2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/226760We characterize the first Alexander Z[Z]-modules of ribbon surface-links in the 4-sphere fixing the number of components and the total genus, and then the first Alexander Z[Z]-modules of surface-links in the 4-sphere fixing the number of components. Using the result of ribbon torus-links, we also characterize the first Alexander Z[Z]-modules of virtual links fixing the number of components. For a general surface-link, an estimate of the total genus is given in terms of the first Alexander Z[Z]-module. We show a graded structure on the first Alexander Z[Z]-modules of all surface-links and then a graded structure on the first Alexander Z[Z]-modules of classical links, surface-links and higher-dimensional manifold-links.This is the version published by Geometry & Topology Monographs on 29 April 2008Geometric Topology57M25, 57Q35, 57Q45text