# 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 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.[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.
• 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 ${\displaystyle A\to A_{P}}$, 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