P-primary. Not all P-primary ideals need be powers of P, but at least they contain a power of P; for example the ideal (x, y2) is P-primary for the ideal P = (x, y) in the ring k[x, y], but is not a power of P, however it contains P². If A is a Noetherian ring and P a prime ideal, then the kernel of , the map from A to the localization of A at P, is the intersection of all P-primary ideals. A finite nonempty product of -primary ideals is -primary but an infinite product of -primary ideals may not be -primary; since for example, in a Noetherian local ring with maximal ideal , (Krull intersection theorem) where each is -primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal of the local ring yields the zero ideal, which in this case is not primary (because the zero divisor is not nilpotent). In fact, in a Noetherian ring, a nonempty product of -primary ideals is -primary if and only if there exists some integer such that . Footnotes References Bourbaki, Algèbre commutative. On primal ideals, Ladislas Fuchs External links Primary ideal at Encyclopaedia of Mathematics Commutative algebra Ideals (ring theory)