  We solved the long-standing problem of describing the cohomology ring of
semiample hypersurfaces in complete simplicial toric varieties. Also, the
monomial-divisor mirror map is generalized to a map between the whole Picard
group and the space of infinitesimal deformations for a mirror pair of
Calabi-Yau hypersurfaces. This map is compatible with certain vanishing
limiting products of the subrings of the chiral rings, on which the ring
structure is related to a product of the roots of $A$-type Lie algebra.
