A primary ideal is a concept in commutative algebra, defined for ideals of a commutative ring with unity.
Definition
Let $R$ be a commutative ring with identity and let $Q$ be a proper ideal of $R$. $Q$ is called primary if for any elements $a, b \in R$,
$$ ab \in Q ;\Longrightarrow; a \in Q \ \text{or}\ b^{n} \in Q \text{ for some } n \ge 1 . $$
Equivalently, an ideal $Q$ is primary precisely when its radical $\sqrt{Q}$ is a prime ideal $P$ and whenever $ab \in Q$ with $a otin Q$, then $b \in P$.
Key Properties
- Radical is prime: If $Q$ is primary, then $\sqrt{Q}$ is a prime ideal. Conversely, if $\sqrt{Q}=P$ is prime and the above “power condition” holds, then $Q$ is primary.
- Zero divisors modulo $Q$: In the quotient ring $R/Q$, every zero divisor is nilpotent. That is, if $\overline{a},\overline{b}=0$ in $R/Q$ and $\overline{a} eq 0$, then $\overline{b}$ is nilpotent.
- P-primary ideals: If $\sqrt{Q}=P$, one often says that $Q$ is $P$-primary.
- Primary decomposition: In Noetherian rings, every proper ideal can be expressed as a finite intersection of primary ideals, a statement known as the Lasker–Noether theorem or primary decomposition. The associated primes of an ideal are the radicals of the primary components in such a decomposition.
Examples
- In the ring $\mathbb{Z}$, the ideal $(p^{k})$ generated by a power of a prime integer $p$ is $ (p)$-primary. Its radical is $(p)$.
- In the polynomial ring $k[x]$ over a field $k$, the ideal $(x^{n})$ is $(x)$-primary.
- In $k[x,y]$, the ideal $(x, y^{2})$ is primary with radical $(x, y)$, which is a maximal (hence prime) ideal.
Relations to Other Concepts
- Prime ideals: Every prime ideal is primary, because the condition reduces to: if $ab\in P$ and $a otin P$, then $b\in P$ (no power needed). Thus prime ideals are exactly the radical primary ideals.
- Nilpotent ideals: The nilradical of a ring (intersection of all prime ideals) is a primary ideal only in trivial cases; generally it is a radical ideal rather than primary.
References
- M. F. Atiyah and I. G. Macdonald, Introduction to Commutative Algebra, Chap. 4.
- D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, §4.1.
- H. Matsumura, Commutative Ring Theory, Chap. 8.
