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 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 .
Properties
All rings are assumed to be commutative and
unital.
Every
proper idealI in a ring has at least one minimal prime ideal above it. The proof of this fact uses
Zorn's lemma.[1] 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.
The
radical of any proper ideal I coincides with the intersection of the minimal prime ideals over I. This follows from the fact that every prime ideal contains a minimal prime ideal.
The set of
zero divisors of a given ring contains the union of the minimal prime ideals.[4]
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.
When is Noetherian
local, with maximal ideal , is minimal over if and only if there exists a number such that .
Equidimensional ring
For a minimal prime ideal in a local ring , in general, it need not be the case that , the
Krull dimension of .
A Noetherian local ring is said to be equidimensional if for each minimal prime ideal , . For example, a local Noetherian
integral domain and a local
Cohen–Macaulay ring are equidimensional.
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 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 .
Properties
All rings are assumed to be commutative and
unital.
Every
proper idealI in a ring has at least one minimal prime ideal above it. The proof of this fact uses
Zorn's lemma.[1] 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.
The
radical of any proper ideal I coincides with the intersection of the minimal prime ideals over I. This follows from the fact that every prime ideal contains a minimal prime ideal.
The set of
zero divisors of a given ring contains the union of the minimal prime ideals.[4]
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.
When is Noetherian
local, with maximal ideal , is minimal over if and only if there exists a number such that .
Equidimensional ring
For a minimal prime ideal in a local ring , in general, it need not be the case that , the
Krull dimension of .
A Noetherian local ring is said to be equidimensional if for each minimal prime ideal , . For example, a local Noetherian
integral domain and a local
Cohen–Macaulay ring are equidimensional.