Minimal prime ideal
In mathematics, especially in the area of algebra known as commutative algebra, certain prime ideals called minimal prime ideals play an important role in understanding rings and modules. The notion of height and Krull's principal ideal theorem use minimal primes.
A prime ideal P is said to be a minimal prime ideal over an ideal I if it is minimal among all prime ideals containing I. (Note: if I is a prime ideal, then I is the only minimal prime over it.) A prime ideal is said to be a minimal prime ideal if it is a minimal prime ideal over the zero ideal.
- In a commutative artinian ring, every maximal ideal is a minimal prime ideal.
- In an integral domain, the only minimal prime ideal is the zero ideal.
- In the ring Z of integers, the minimal prime ideals over a nonzero principal ideal (n) are the principal ideals (p), where p is a prime divisor of n. The only minimal prime ideal over the zero ideal is the zero ideal itself. Similar statements hold for any principal ideal domain.
- If I is a p-primary ideal (for example, a symbolic power of p), then p is the unique minimal prime ideal over I.
- The ideals and are the minimal prime ideals in since they are the extension of prime ideals for the morphism , contain the zero ideal (which is not prime since , but, neither nor are contained in the zero ideal) and are not contained in any other prime ideal.
- In the minimal primes over the ideal are the ideals and .
- Let and the images of x, y in A. Then and are the minimal prime ideals of A (and there are no others). Let be the set of zero-divisors in A. Then is in D (since it kills nonzero ) while neither in nor ; so .
All rings are assumed to be commutative and unital.
- Every proper ideal I in a ring has at least one minimal prime ideal above it. The proof of this fact uses Zorn's lemma. Any maximal ideal containing I is prime, and such ideals exist, so the set of prime ideals containing I is non-empty. The intersection of a decreasing chain of prime ideals is prime. Therefore, the set of prime ideals containing I has a minimal element, which is a minimal prime over I.
- Emmy Noether showed that in a Noetherian ring, there are only finitely many minimal prime ideals over any given ideal. The fact remains true if "Noetherian" is replaced by the ascending chain conditions on radical ideals.
- The radical of any proper ideal I coincides with the intersection of the minimal prime ideals over I.
- The set of zero divisors of a given ring contains the union of the minimal prime ideals.
- Krull's principal ideal theorem says that, in a Noetherian ring, each minimal prime over a principal ideal has height at most one.
- Each proper ideal I of a Noetherian ring contains a product of the possibly repeated minimal prime ideals over it (Proof: is the intersection of the minimal prime ideals over I. For some n, and so I contains .)
- A prime ideal in a ring R is a unique minimal prime over an ideal I if and only if , and such an I is -primary if is maximal. This gives a local criterion for a minimal prime: a prime ideal is a minimal prime over I if and only if is a -primary ideal. When R is a Noetherian ring, is a minimal prime over I if and only if is an Artinian ring (i.e., is nilpotent module I). The pre-image of under is a primary ideal of called the -primary component of I.
For a minimal prime ideal in a local ring , in general, it need not be the case that , the Krull dimension of .