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Inseparability criteria for continuous and discrete bipartite quantum statesbased on moments of annihilation and creation operators are studied bydeveloping the idea of Shchukin-Vogel criterion [Phys. Rev. Lett. {\bf 95},230502 (2005)]. If a state is separable, then the corresponding matrix ofmoments is separable too. Thus, we derive generalized criteria, based on theseparability properties of the matrix of moments, are thus derived. Inparticular, a new criterion based on realignment of moments in the matrix isproposed as an analogue of the standard realignment criterion for densitymatrices. Other inseparability inequalities are obtained by applying positivemaps to the matrix of moments. Usefulness of the Shchukin-Vogel criterion todescribe bipartite-entanglement of more than two modes is demonstrated: Weobtain some previously known three-mode inseparability criteria originallyderived from the Cauchy-Schwarz inequality, and we introduce new ones.Comment: 11 pages; shortened version, introduction improved, accepted for  publication in PR

Inseparability criteria based on matrices of moments