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 $\mathfrak{q}$ is primary if, whenever $x y \in \mathfrak{q}$, we have either $x \in \mathfrak{q}$ or $y \in \mathfrak{q}$ or $x, y \in \sqrt{\mathfrak{q}}$. (Here $\sqrt{\mathfrak{q}}$ denotes the radical of $\mathfrak{q}$.)
• 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 $R = k[x,y,z]/(x y - z^2)$, $\mathfrak{p} = (\overline{x}, \overline{z})$, 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.
• 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 $A \to A_P$, 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

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

External links

• Primary ideal at Encyclopaedia of Mathematics