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: a proper ideal ${\displaystyle {\mathfrak {q))}$ is primary if, whenever ${\displaystyle xy\in {\mathfrak {q))}$, we have ${\displaystyle x\in {\mathfrak {q))}$ or ${\displaystyle y\in {\mathfrak {q))}$ or ${\displaystyle x,y\in {\sqrt {\mathfrak {q))))$. (Here ${\displaystyle {\sqrt {\mathfrak {q))))$ denotes the radical of ${\displaystyle {\mathfrak {q))}$.)
• 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.^[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.
    • 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.
• 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 ${\displaystyle A\to A_{P))$, the map from A to the localization of A at P, is the intersection of all P-primary ideals.^[4]
• A finite nonempty product of ${\displaystyle {\mathfrak {p))}$-primary ideals is ${\displaystyle {\mathfrak {p))}$-primary but an infinite product of ${\displaystyle {\mathfrak {p))}$-primary ideals may not be ${\displaystyle {\mathfrak {p))}$-primary; since for example, in a Noetherian local ring with maximal ideal ${\displaystyle {\mathfrak {m))}$, ${\displaystyle \cap _{n>0}{\mathfrak {m))^{n}=0}$ (Krull intersection theorem) where each ${\displaystyle {\mathfrak {m))^{n))$ is ${\displaystyle {\mathfrak {m))}$-primary, for example the infinite product of the maximal (and hence prime and hence primary) ideal ${\displaystyle m=\langle x,y\rangle }$ of the local ring ${\displaystyle K[x,y]/\langle x^{2},xy\rangle }$ yields the zero ideal, which in this case is not primary (because the zero divisor ${\displaystyle y}$ is not nilpotent). In fact, in a Noetherian ring, a nonempty product of ${\displaystyle {\mathfrak {p))}$-primary ideals ${\displaystyle Q_{i))$ is ${\displaystyle {\mathfrak {p))}$-primary if and only if there exists some integer ${\displaystyle n>0}$ such that ${\displaystyle {\mathfrak {p))^{n}\subset \cap _{i}Q_{i))$.^[5]