Primary ideal
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In mathematics, an ideal Q of a commutative ring A is said to be primary if when 
either 
or 
for some n. Every prime ideal is primary.
For example, in 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 noetherian ring has a primary decomposition: that is, an ideal can be written as an intersection of finitely many primary ideals. (This result is known as Lasker–Noether theorem.) Consequently,[1] an irreducible ideal of a noetherian ring is primary.
Every prime ideal is primary and every primary ideal is primal.
If Q is a primary ideal, then the associated prime ideal P is the radical of Q. If P is the radical of the primary ideal Q, then Q is said to be is P-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.
[edit] Footnotes
- ^ To be precise, one usually uses this fact to prove the theorem.
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