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 $xy\in {\mathfrak {q}}$, 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 $R=k[x,y,z]/(xy-z^{2})$, ${\mathfrak {p}}=(\overline {x},\overline {z})$, 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.
• 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