# Primary ideal

In mathematics, specifically commutative algebra, a proper ideal Q of a commutative ring A is said to be primary if whenever xy is an element of Q then x or yn is also an element of Q, for some n>0. For example, in the ring of integers Z, (pn) is a primary ideal if p is a prime number.

The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition, that is, can be written as an intersection of finitely many primary ideals. This result is known as the Lasker–Noether theorem. Consequently,[1] an irreducible ideal of a Noetherian ring is primary.

Various methods of generalizing primary ideals to noncommutative rings exist[2] but the topic is most often studied for commutative rings. Therefore, the rings in this article are assumed to be commutative rings with identity.

## Examples and properties

• The definition can be rephrased in a more symmetric manner: an ideal is primary if, whenever , we have either or or . (Here denotes the radical of .)
• An ideal Q of R is primary if and only if every zerodivisor in R/Q is nilpotent. (Compare this to the case of prime ideals, where P is prime if every zerodivisor in R/P is actually zero.)
• 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.[3]
• 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.
• 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 (x, y2) is P-primary for the ideal P = (x, y) in the ring k[x, y], but is not a power of P.
• In general powers of a prime ideal P need not be P-primary. (An example is given by taking R to be the ring k[x, y, z]/(xy  z2), with P the prime ideal (x, z). If Q = P2, then xy  Q, but x is not in Q and y is not in the radical P of Q, so Q is not P-primary.) However every ideal Q with radical P is contained in a smallest P-primary ideal, consisting of all elements a such that ax is in Q for some x not in P. In particular there is a smallest P-primary ideal containing Pn, called the nth symbolic 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.[4]

## Footnotes

1. To be precise, one usually uses this fact to prove the theorem.
2. See the references to Chatters-Hajarnavis, Goldman, Gorton-Heatherly, and Lesieur-Croisot.
3. For the proof of the second part see the article of Fuchs
4. Atiyah-Macdonald, Corollary 10.21