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, 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: 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.

• 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.

References

• On primal ideals, Ladislas Fuchs

External links

• Primary ideal at Encyclopaedia of Mathematics