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, an irreducible ideal of a Noetherian ring is primary.
Various methods of generalizing primary ideals to noncommutative rings exist 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.
- 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.
- To be precise, one usually uses this fact to prove the theorem.
- See the references to Chatters-Hajarnavis, Goldman, Gorton-Heatherly, and Lesieur-Croisot.
- For the proof of the second part see the article of Fuchs
- Atiyah-Macdonald, Corollary 10.21
- Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, p. 50, ISBN 978-0-201-40751-8
- Goldman, Oscar (1969), Rings and modules of quotients, J. Algebra 13: 10–47, doi:10.1016/0021-8693(69)90004-0, ISSN 0021-8693, MR 0245608
- Gorton, Christine; Heatherly, Henry (2006), Generalized primary rings and ideals, Math. Pannon. 17 (1): 17–28, ISSN 0865-2090, MR 2215638
- On primal ideals, Ladislas Fuchs
- Lesieur, L.; Croisot, R. (1963), Algèbre noethérienne non commutative (in French), Mémor. Sci. Math., Fasc. CLIV. Gauthier-Villars & Cie, Editeur -Imprimeur-Libraire, Paris, p. 119, MR 0155861