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 either ${\displaystyle x\in {\mathfrak {q}}}$ 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.
• 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 axQ for some xP. 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; 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.
• 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]

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