The algebra of integro-differential operators on a polynomial algebra

We prove that the algebra I-n := K < x(1), ... , x(n), ... x(1), partial derivative/partial derivative x(1), ... , partial derivative/partial derivative x(n), integral(1), ... , integral(n) > of integro-differential operators on a polynomial algebra is a prime, central, catenary, self-dual, non-Noetherian algebra of classical Krull dimension n and of Gelfand-Kirillov dimension 2n. Its weak homological dimension is n, and n <= gldim(I-n) <= 2n. All the ideals of I-n are found explicitly, there are only finitely many of them (at most 2(2n)), they commute (ab - ba) and are idempotent ideals (a(2) = a). The number of ideals of I-n is equal to the Dedekind number partial derivative(n). An analogue of Hilbert's Syzygy Theorem is proved for I-n. The group of units of the algebra In is described (it is a huge group). A canonical form is found for each integro-differential operator (by proving that the algebra I-n is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra A(n) (but GK(A(n)) = 3n, note that I-n subset of A(n)). It is proved that the algebras I-n and A(n) are ideal equivalent.