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In contrast to its subalgebra  of polynomial differential operators (i.e. the ’th Weyl algebra), the algebra   of polynomial integro-differential operators is neither left nor right Noetherian algebra; moreover it contains infinite direct sums of nonzero left and right ideals. It is proved that  is a left (right) coherent algebra iff  = 1; the algebra  is a    of length 3 and has multiplicity 3 with respect to the filtration of Bernstein, and all 3 simple factors of  are pairwise non-isomorphic  -bimodules. The socle length of the  -bimodule  is  + 1, the socle filtration is found, and the ’th term of the socle filtration has length . This fact gives a new canonical form for each polynomial integro-differential operator. It is proved that the algebra  is the maximal left (resp. right) order in the largest left (resp. right) quotient ring of the algebra .

The Algebra of Polynomial Integro-Differential Operators is a Holonomic Bimodule over the Subalgebra of Polynomial Differential Operators