We study algebras of modular forms on unitary groups of signature $(n, 1)$. We give a sufficient criterion for the ring of unitary modular forms to be freely generated in terms of the divisor of a modular Jacobian determinant. We use this to prove that a number of rings of unitary modular forms associated to Hermitian lattices over the rings of integers of ${Q}(\sqrt{ d})$ for $d = -1, -2, -3$ are polynomial algebras without relations. This is joint work with Haowu Wang.