An interesting identity is obtained for ideals A and B in a Noetherian ring :

A = (A + Bn ) ∩ (A : Bn )

for sufficiently large n. This identity is applied to obtain Fuchs' quasi-primary decomposition of A in an improved form, and to obtain Krull's theorem on the intersection of the powers of A, both developments making no use of the Noetherian primary decomposition of A. Finally, the identity is used to obtain the primary decomposition without reference to irreducible ideals, in a largely constructive manner which yields the decomposition in an illuminating, automatically normal form, and which, subject to certain simple conditions, is unique.