Minimal prime ideal

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. A minimal prime ideal over an ideal I in a Noetherian ring R is precisely a minimal associated prime (also called isolated prime) of R/I; this follows for instance from the primary decomposition of I. ==Examples==

• 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 (x) and (y) are the minimal prime ideals in \mathbb{C}[x,y]/(xy) since they are the extension of prime ideals for the morphism \mathbb{C}[x,y] \to \mathbb{C}[x,y]/(xy), contain the zero ideal (which is not prime since x\cdot y = 0 \in (0), but, neither x nor y are contained in the zero ideal) and are not contained in any other prime ideal. • In \mathbb{C}[x,y,z] the minimal primes over the ideal ((x^3 - y^3 - z^3)^4 (x^5 + y^5 + z^5)^3) are the ideals (x^3 - y^3 - z^3) and (x^5 + y^5 + z^5). • Let A = \mathbb{C}[x,y]/(x^3 y, x y^3) and \overline{x}, \overline{y} the images of x, y in A. Then (\overline{x}) and (\overline{y}) are the minimal prime ideals of A (and there are no others). Let D be the set of zero-divisors in A. Then \overline{x} + \overline{y} is in D (since it kills nonzero \overline{x}^2 \overline{y} - \overline{x}\overline{y}^2) while neither in (\overline{x}) nor (\overline{y}); so (\overline{x}) \cup (\overline{y}) \subsetneq D. ==Properties==

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 \sqrt{I} 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. • 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: \sqrt{I} = \bigcap_i^r \mathfrak{p}_i is the intersection of the minimal prime ideals over I. For some n, \sqrt{I}^n \subset I and so I contains \prod_1^r \mathfrak{p}_i^n.) • A prime ideal \mathfrak{p} in a ring R is a unique minimal prime over an ideal I if and only if \sqrt{I} = \mathfrak{p}, and such an I is \mathfrak{p}-primary if \mathfrak{p} is maximal. This gives a local criterion for a minimal prime: a prime ideal \mathfrak{p} is a minimal prime over I if and only if I R_{\mathfrak{p}} is a \mathfrak{p} R_{\mathfrak{p}}-primary ideal. When R is a Noetherian ring, \mathfrak{p} is a minimal prime over I if and only if R_{\mathfrak{p}}/I R_{\mathfrak{p}} is an Artinian ring (i.e., \mathfrak{p} R_{\mathfrak{p}} is nilpotent module I). The pre-image of I R_{\mathfrak{p}} under R \to R_{\mathfrak{p}} is a primary ideal of R called the \mathfrak{p}-primary component of I. • When A is Noetherian local, with maximal ideal P, P\supseteq I is minimal over I if and only if there exists a number m such that P^m\subseteq I. == Equidimensional ring ==

For a minimal prime ideal \mathfrak{p} in a local ring A, in general, it need not be the case that \dim A/\mathfrak{p} = \dim A, the Krull dimension of A. A Noetherian local ring A is said to be equidimensional if for each minimal prime ideal \mathfrak{p}, \dim A/\mathfrak{p} = \dim A. For example, a local Noetherian integral domain and a local Cohen–Macaulay ring are equidimensional. See also equidimensional scheme and quasi-unmixed ring. == See also ==