Primary ideal

In mathematics, specifically commutative algebra, a proper ideal ${\displaystyle {\mathfrak {q}}}$ of a commutative ring ${\displaystyle A}$ is said to be primary if whenever ${\displaystyle xy}$ is an element of ${\displaystyle {\mathfrak {q}}}$ then ${\displaystyle x}$ or ${\displaystyle y^{n}}$ is also an element of ${\displaystyle {\mathfrak {q}}}$, for some ${\displaystyle n>0}$. For example, in the ring of integers ${\displaystyle \mathbb {Z} }$, ${\displaystyle (p^{n})}$ is a primary ideal if ${\displaystyle p}$ 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.

• The definition can be rephrased in a more apparently symmetrical manner: a proper ideal ${\displaystyle {\mathfrak {q}}}$ is primary if, whenever ${\displaystyle xy\in {\mathfrak {q}}}$, ${\displaystyle x}$ or ${\displaystyle y}$ are elements of ${\displaystyle {\mathfrak {q}}}$, or both ${\displaystyle x}$ and ${\displaystyle y}$ lie in ${\displaystyle {\sqrt {\mathfrak {q}}}}$, the radical of ${\displaystyle {\mathfrak {q}}}$; i.e., ${\displaystyle xy\in {\mathfrak {q}}\implies (x\in {\mathfrak {q}})\lor (y\in {\mathfrak {q}})\lor ((x\in {\sqrt {\mathfrak {q}}})\land (y\in {\sqrt {\mathfrak {q}}})).}$
• A proper ideal ${\displaystyle {\mathfrak {q}}}$ of ${\displaystyle R}$ is primary if and only if every zero divisor in ${\displaystyle R/{\mathfrak {q}}}$ is nilpotent. (Compare this to the case of prime ideals, where ${\displaystyle {\mathfrak {p}}}$ is prime if and only if every zero divisor in ${\displaystyle R/{\mathfrak {p}}}$ 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 ${\displaystyle {\mathfrak {q}}}$ is a primary ideal, then the radical of ${\displaystyle {\mathfrak {q}}}$ is necessarily a prime ideal ${\displaystyle {\mathfrak {p}}}$, and this ideal is called the associated prime ideal of ${\displaystyle {\mathfrak {q}}}$. In this situation, ${\displaystyle {\mathfrak {q}}}$ is said to be ${\displaystyle {\mathfrak {p}}}$-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 ${\displaystyle n>0}$, 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 ${\displaystyle {\mathfrak {q}}}$ with radical ${\displaystyle {\mathfrak {p}}}$ is contained in a smallest ${\displaystyle {\mathfrak {p}}}$-primary ideal: all elements ${\displaystyle a}$ such that ${\displaystyle ax\in {\mathfrak {q}}}$ for some ${\displaystyle x\notin {\mathfrak {p}}}$. The smallest ${\displaystyle {\mathfrak {p}}}$-primary ideal containing ${\displaystyle {\mathfrak {p}}^{n}}$ is called the ${\displaystyle n}$th symbolic power of ${\displaystyle {\mathfrak {p}}}$.
• If ${\displaystyle {\mathfrak {p}}}$ is a maximal prime ideal, then any ideal whose radical is ${\displaystyle {\mathfrak {p}}}$ is ${\displaystyle {\mathfrak {p}}}$-primary (and vice versa). In particular, a power of ${\displaystyle {\mathfrak {p}}}$ or an ideal containing a power of ${\displaystyle {\mathfrak {p}}}$ is ${\displaystyle {\mathfrak {p}}}$-primary. But a ${\displaystyle {\mathfrak {p}}}$-primary ideal need not be a power of ${\displaystyle {\mathfrak {p}}}$ and need not contain a power of ${\displaystyle {\mathfrak {p}}}$; for example, the ideal ${\displaystyle (x,y^{2})}$ is ${\displaystyle {\mathfrak {p}}}$-primary for the ideal ${\displaystyle {\mathfrak {p}}=(x,y)}$ in the ring ${\displaystyle k[x,y]}$, but is not a power of ${\displaystyle {\mathfrak {p}}}$; however, it contains ${\displaystyle {\mathfrak {p}}^{2}}$.
• If ${\displaystyle A}$ is a Noetherian ring and ${\displaystyle {\mathfrak {p}}}$ a prime ideal, then the kernel of ${\displaystyle A\to A_{\mathfrak {p}}}$, the map from ${\displaystyle A}$ to the localization of ${\displaystyle A}$ at ${\displaystyle {\mathfrak {p}}}$, is the intersection of all ${\displaystyle {\mathfrak {p}}}$-primary ideals.⁴
• If ${\displaystyle {\mathfrak {p}}}$ is maximal, 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 ${\displaystyle {\mathfrak {m}}}$, ${\displaystyle \textstyle \bigcap _{n>0}{\mathfrak {m}}^{n}=0}$ (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 ${\displaystyle {\mathfrak {p}}^{n}\subset \textstyle \bigcap _{i}Q_{i}}$.⁵

The primary decomposition of ideals by the Lasker–Noether theorem may be seen as a generalization of the fundamental theorem of arithmetic, which applies to the integers ${\displaystyle \mathbb {Z} }$ and other unique factorization domains, to general Noetherian rings. While the unique factorization of elements of a ring into the product of irreducible elements (up to units and reordering) fails in the general case, the Lasker–Noether theorem states that the ideals of a Noetherian ring do still have a type of "unique factorization": any ideal in a Noetherian ring can be written as an intersection of primary ideals of the ring in a primary decomposition, and while these component primary ideals are not necessarily unique, the radicals of these components, the associated primes of the ideal, are unique up to reordering: