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: a proper ideal ${\mathfrak {q}}$ is primary if, whenever $xy\in {\mathfrak {q}}$ , we have $x\in {\mathfrak {q}}$ or $y\in {\mathfrak {q}}$ or $x,y\in {\sqrt {\mathfrak {q}}}$ . (Here ${\sqrt {\mathfrak {q}}}$ denotes the radical of ${\mathfrak {q}}$ .)
A proper 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 $R=k[x,y,z]/(xy-z^{2})$ , ${\mathfrak {p}}=({\overline {x}},{\overline {z}})$ , and ${\mathfrak {q}}={\mathfrak {p}}^{2}$ , then ${\mathfrak {p}}$ is prime and ${\sqrt {\mathfrak {q}}}={\mathfrak {p}}$ , but we have ${\overline {x}}{\overline {y}}={\overline {z}}^{2}\in {\mathfrak {p}}^{2}={\mathfrak {q}}$ , ${\overline {x}}\not \in {\mathfrak {q}}$ , and ${\overline {y}}^{n}\not \in {\mathfrak {q}}$ for all n > 0, so ${\mathfrak {q}}$ is not primary. The primary decomposition of ${\mathfrak {q}}$ is $({\overline {x}})\cap ({\overline {x}}^{2},{\overline {x}}{\overline {z}},{\overline {y}})$ ; here $({\overline {x}})$ is ${\mathfrak {p}}$ -primary and $({\overline {x}}^{2},{\overline {x}}{\overline {z}},{\overline {y}})$ is $({\overline {x}},{\overline {y}},{\overline {z}})$ -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 (x, y²) is P-primary for the ideal P = (x, y) in the ring k[x, y], 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 $A\to A_{P}$ , 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 ${\mathfrak {p}}$ -primary ideals is ${\mathfrak {p}}$ -primary but an infinite product of ${\mathfrak {p}}$ -primary ideals may not be ${\mathfrak {p}}$ -primary; since for example, in a Noetherian local ring with maximal ideal ${\mathfrak {m}}$ , $\cap _{n>0}{\mathfrak {m}}^{n}=0$ (Krull intersection theorem) where each ${\mathfrak {m}}^{n}$ is ${\mathfrak {m}}$ -primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal $m=\langle x,y\rangle$ of the local ring $K[x,y]/\langle x^{2},xy\rangle$ yields the zero ideal, which in this case is not primary (because the zero divisor $y$ is not nilpotent). In fact, in a Noetherian ring, a nonempty product of ${\mathfrak {p}}$ -primary ideals $Q_{i}$ is ${\mathfrak {p}}$ -primary if and only if there exists some integer $n>0$ such that ${\mathfrak {p}}^{n}\subset \cap _{i}Q_{i}$ .^[5]

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
^ 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