# 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 *y*^{n} is also an element of *Q*, for some *n>0*. For example, in the ring of integers **Z**, (*p*^{n}) 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*,*y*^{2}) 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*−*z*^{2}), with*P*the prime ideal (*x*,*z*). If*Q*=*P*^{2}, 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*P*^{n}, called the*n*th**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

- ↑ 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

## References

- Atiyah, Michael Francis; Macdonald, I.G. (1969),
*Introduction to Commutative Algebra*, Westview Press, p. 50, ISBN 978-0-201-40751-8 - Chatters, A. W.; Hajarnavis, C. R. (1971), "Non-commutative rings with primary decomposition",
*Quart. J. Math. Oxford Ser. (2)*,**22**: 73–83, doi:10.1093/qmath/22.1.73, ISSN 0033-5606, MR 0286822 - 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