Primary ideal

Examples and properties

• The definition can be rephrased in a more symmetric manner: an ideal is primary if, whenever, we have or or.
• An ideal Q of R is primary if and only if every zero divisor in R/Q is nilpotent.
• Any prime ideal is primary, and moreover an ideal is prime if and only if it is primary and semiprime.
• Every primary ideal is primal.
• If Q is a primary ideal, then the radical of Q is necessarily a prime ideal P, and this ideal is called the associated prime ideal of Q. In this situation, Q is said to be P-primary.
• * On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if,, and, then is prime and, but we have,, and for all n > 0, so is not primary. The primary decomposition of is ; here is -primary and is -primary.
• ** An ideal whose radical is maximal, however, is primary.
• ** Every ideal with radical is contained in a smallest -primary ideal: all elements such that for some. The smallest -primary ideal containing is called the th symbolic power of.
• If P is a maximal prime ideal, then any ideal containing a power of P is P-primary. Not all P-primary ideals need be powers of P; for example the ideal is P-primary for the ideal P = in the ring k, but is not a power of P.
• 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, where each is -primary. 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