We 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.
