  We determine the structure of the bigraded ring of weak Jacobi forms with
integral Fourier coefficients. This ring is the target ring of a map
generalising the Witten and elliptic genera and a partition function of
$(0,2)$-model in string theory. We also determine the structure of the graded
ring of all weakly holomorphic Jacobi forms of weight zero and integral index
with integral Fourier coefficients. These forms are the data for Borcherds
products for the Siegel paramodular groups.
