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We prove that the algebra  n   :  =  K   〈x  1   ,  …  ,   x  n   ,    ∂   ∂   x  1,  …  ,    ∂   ∂   x  n,∫  1    ,  …  ,∫  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 2 n . Its weak homological dimension is  n , and  n , ⩽ gldim(  n  ) ⩽ 2 n . All the ideals of   n   are found explicitly, there are only finitely many of them (at most2    2   n), they commute (= ) and are idempotent ideals ( 2  = ). The number of ideals of   n   is equal to the  Dedekind number    n  . An analogue of Hilbert's Syzygy Theorem is proved for   n  . The group of units of the algebra   n   is described (it is a huge group). A canonical form is found for each integro‐differential operator (by proving that the algebra   n   is a generalized Weyl algebra). All the mentioned results hold for the Jacobian algebra   n   (but GK (  n  ) = 3 n , note that   n   ⊂   n  ). It is proved that the algebras   n   and   n   are ideal equivalent.

The algebra of integro‐differential operators on a polynomial algebra