Primary ideal

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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[edit]

  • 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.
  • 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 (xy2) is P-primary for the ideal P = (xy) in the ring k[xy], 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[xyz]/(xy − z2), with P the prime ideal (xz). 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[edit]

  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

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