
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, an irreducible ideal of a Noetherian ring is primary.
Various methods of generalizing primary ideals to noncommutative rings exist, 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
is primary if, whenever
, we have
or
or
. (Here
denotes the radical of
.)
- 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.
- 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
,
, and
, then
is prime and
, but we have
,
, and
for all n > 0, so
is not primary. The primary decomposition of
is
; here
is
-primary and
is
-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.
- On the other hand, an ideal whose radical is prime is not necessarily primary: for example, if
- 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
, the map from A to the localization of A at P, is the intersection of all P-primary ideals.
- A finite nonempty product of
-primary ideals is
-primary but an infinite product of
-primary ideals may not be
-primary; since for example, in a Noetherian local ring with maximal ideal
,
(Krull intersection theorem) where each
is
-primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal
of the local ring
yields the zero ideal, which in this case is not primary (because the zero divisor
is not nilpotent). In fact, in a Noetherian ring, a nonempty product of
-primary ideals
is
-primary if and only if there exists some integer
such that
.
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
External links
- Primary ideal at Encyclopaedia of Mathematics
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 gt 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 an irreducible ideal of a Noetherian ring is primary Various methods of generalizing primary ideals to noncommutative rings exist 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 propertiesThe definition can be rephrased in a more symmetric manner a proper ideal q displaystyle mathfrak q is primary if whenever xy q displaystyle xy in mathfrak q we have x q displaystyle x in mathfrak q or y q displaystyle y in mathfrak q or x y q displaystyle x y in sqrt mathfrak q Here q displaystyle sqrt mathfrak q denotes the radical of q displaystyle 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 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 z2 displaystyle R k x y z xy z 2 p x z displaystyle mathfrak p overline x overline z and q p2 displaystyle mathfrak q mathfrak p 2 then p displaystyle mathfrak p is prime and q p displaystyle sqrt mathfrak q mathfrak p but we have x y z 2 p2 q displaystyle overline x overline y overline z 2 in mathfrak p 2 mathfrak q x q displaystyle overline x not in mathfrak q and y n q displaystyle overline y n not in mathfrak q for all n gt 0 so q displaystyle mathfrak q is not primary The primary decomposition of q displaystyle mathfrak q is x x 2 x z y displaystyle overline x cap overline x 2 overline x overline z overline y here x displaystyle overline x is p displaystyle mathfrak p primary and x 2 x z y displaystyle overline x 2 overline x overline z overline y is x y z 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 n th 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 P If A is a Noetherian ring and P a prime ideal then the kernel of A AP displaystyle A to A P the map from A to the localization of A at P is the intersection of all P primary ideals A finite nonempty product of p displaystyle mathfrak p primary ideals is p displaystyle mathfrak p primary but an infinite product of p displaystyle mathfrak p primary ideals may not be p displaystyle mathfrak p primary since for example in a Noetherian local ring with maximal ideal m displaystyle mathfrak m n gt 0mn 0 displaystyle cap n gt 0 mathfrak m n 0 Krull intersection theorem where each mn displaystyle mathfrak m n is m displaystyle mathfrak m primary for example the infinite product of the maximal and hence prime and hence primary ideal m x y displaystyle m langle x y rangle of the local ring K x y x2 xy displaystyle K x y langle x 2 xy rangle yields the zero ideal which in this case is not primary because the zero divisor y displaystyle y is not nilpotent In fact in a Noetherian ring a nonempty product of p displaystyle mathfrak p primary ideals Qi displaystyle Q i is p displaystyle mathfrak p primary if and only if there exists some integer n gt 0 displaystyle n gt 0 such that pn iQi displaystyle mathfrak p n subset cap i Q i FootnotesTo 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 ReferencesAtiyah Michael Francis Macdonald I G 1969 Introduction to Commutative Algebra Westview Press p 50 ISBN 978 0 201 40751 8 Bourbaki Algebre 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 Algebre noetherienne non commutative in French Memor Sci Math Fasc CLIV Gauthier Villars amp Cie Editeur Imprimeur Libraire Paris p 119 MR 0155861External linksPrimary ideal at Encyclopaedia of Mathematics