  Inseparability criteria for continuous and discrete bipartite quantum states
based on moments of annihilation and creation operators are studied by
developing the idea of Shchukin-Vogel criterion [Phys. Rev. Lett. {\bf 95},
230502 (2005)]. If a state is separable, then the corresponding matrix of
moments is separable too. Thus, we derive generalized criteria, based on the
separability properties of the matrix of moments, are thus derived. In
particular, a new criterion based on realignment of moments in the matrix is
proposed as an analogue of the standard realignment criterion for density
matrices. Other inseparability inequalities are obtained by applying positive
maps to the matrix of moments. Usefulness of the Shchukin-Vogel criterion to
describe bipartite-entanglement of more than two modes is demonstrated: We
obtain some previously known three-mode inseparability criteria originally
derived from the Cauchy-Schwarz inequality, and we introduce new ones.
