# 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 ${\displaystyle {\mathfrak {q}}}$  is primary if, whenever ${\displaystyle xy\in {\mathfrak {q}}}$ , we have ${\displaystyle x\in {\mathfrak {q}}}$  or ${\displaystyle y\in {\mathfrak {q}}}$  or ${\displaystyle x,y\in {\sqrt {\mathfrak {q}}}}$ . (Here ${\displaystyle {\sqrt {\mathfrak {q}}}}$  denotes the radical of ${\displaystyle {\mathfrak {q}}}$ .)
• An ideal Q of R is primary if and only if every zero divisor in R/Q is nilpotent. (Compare this to the case of prime ideals, where P is prime if and only if every zero divisor 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 (also called radical ideal in the commutative case).
• 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 ${\displaystyle R=k[x,y,z]/(xy-z^{2})}$ , ${\displaystyle {\mathfrak {p}}=({\overline {x}},{\overline {z}})}$ , and ${\displaystyle {\mathfrak {q}}={\mathfrak {p}}^{2}}$ , then ${\displaystyle {\mathfrak {p}}}$  is prime and ${\displaystyle {\sqrt {\mathfrak {q}}}={\mathfrak {p}}}$ , but we have ${\displaystyle {\overline {x}}{\overline {y}}={\overline {z}}^{2}\in {\mathfrak {p}}^{2}={\mathfrak {q}}}$ , ${\displaystyle {\overline {x}}\not \in {\mathfrak {q}}}$ , and ${\displaystyle {\overline {y}}^{n}\not \in {\mathfrak {q}}}$  for all n > 0, so ${\displaystyle {\mathfrak {q}}}$  is not primary. The primary decomposition of ${\displaystyle {\mathfrak {q}}}$  is ${\displaystyle ({\overline {x}})\cap ({\overline {x}}^{2},{\overline {x}}{\overline {z}},{\overline {y}})}$ ; here ${\displaystyle ({\overline {x}})}$  is ${\displaystyle {\mathfrak {p}}}$ -primary and ${\displaystyle ({\overline {x}}^{2},{\overline {x}}{\overline {z}},{\overline {y}})}$  is ${\displaystyle ({\overline {x}},{\overline {y}},{\overline {z}})}$ -primary.
• An ideal whose radical is maximal, however, is primary.
• Every ideal Q with radical P is contained in a smallest P-primary ideal: all elements a such that ax ∈ Q for some x ∉ P. The smallest P-primary ideal containing Pn is called the nth symbolic power of P.
• 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, but at least they contain a power of P; for example the ideal (xy2) is P-primary for the ideal P = (xy) in the ring k[xy], 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 ${\displaystyle A\to A_{P}}$ , the map from A to the localization of A at P, is the intersection of all P-primary ideals.[4]
• A finite nonempty product of ${\displaystyle {\mathfrak {p}}}$ -primary ideals is ${\displaystyle {\mathfrak {p}}}$ -primary but an infinite product of ${\displaystyle {\mathfrak {p}}}$ -primary ideals may not be ${\displaystyle {\mathfrak {p}}}$ -primary; since for example, in a Noetherian local ring with maximal ideal ${\displaystyle {\mathfrak {m}}}$ , ${\displaystyle \cap _{n>0}{\mathfrak {m}}^{n}=0}$  (Krull intersection theorem) where each ${\displaystyle {\mathfrak {m}}^{n}}$  is ${\displaystyle {\mathfrak {m}}}$ -primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal ${\displaystyle m=\langle x,y\rangle }$  of the local ring ${\displaystyle K[x,y]/\langle x^{2},xy\rangle }$  yields the zero ideal, which in this case is not primary (because the zero divisor ${\displaystyle y}$  is not nilpotent). In fact, in a Noetherian ring, a nonempty product of ${\displaystyle {\mathfrak {p}}}$ -primary ideals ${\displaystyle Q_{i}}$  is ${\displaystyle {\mathfrak {p}}}$ -primary if and only if there exists some integer ${\displaystyle n>0}$  such that ${\displaystyle {\mathfrak {p}}^{n}\subset \cap _{i}Q_{i}}$ .[5]

## 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
5. ^ Bourbaki, Ch. IV, § 2, Exercise 3.

## References

• Atiyah, Michael Francis; Macdonald, I.G. (1969), Introduction to Commutative Algebra, Westview Press, p. 50, ISBN 978-0-201-40751-8
• Bourbaki, Algèbre commutative.
• Chatters, A. W.; Hajarnavis, C. R. (1971), "Non-commutative rings with primary decomposition", The Quarterly Journal of Mathematics, Second Series, 22: 73–83, doi:10.1093/qmath/22.1.73, ISSN 0033-5606, MR 0286822
• Goldman, Oscar (1969), "Rings and modules of quotients", Journal of 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", Mathematica Pannonica, 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