  We introduce differential primary decompositions for ideals in a commutative
ring. Ideal membership is characterized by differential conditions. The minimal
number of conditions needed is the arithmetic multiplicity. Minimal
differential primary decompositions are unique up to change of bases. Our
results generalize the construction of Noetherian operators for primary ideals
in the analytic theory of Ehrenpreis-Palamodov, and they offer a concise method
for representing affine schemes. The case of modules is also addressed. We
implemented an algorithm in Macaulay2 that computes the minimal decomposition
for an ideal in a polynomial ring.
