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 ${\displaystyle R/I}$; this follows for instance from the primary decomposition of I.

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

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.[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.
• Emmy Noether showed that in a Noetherian ring, there are only finitely many minimal prime ideals over any given ideal.[2][3] The fact remains true if "Noetherian" is replaced by the ascending chain conditions on radical ideals.
• The radical ${\displaystyle {\sqrt {I}}}$ of any proper ideal I coincides with the intersection of the minimal prime ideals over I.[4]
• The set of zero divisors of a given ring contains the union of the minimal prime ideals.[5]
• 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: ${\displaystyle {\sqrt {I}}=\bigcap _{i}^{r}{\mathfrak {p}}_{i}}$ is the intersection of the minimal prime ideals over I. For some n, ${\displaystyle {\sqrt {I}}^{n}\subset I}$ and so I contains ${\displaystyle \prod _{1}^{r}{\mathfrak {p}}_{i}^{n}}$.)
• A prime ideal ${\displaystyle {\mathfrak {p}}}$ in a ring R is a unique minimal prime over an ideal I if and only if ${\displaystyle {\sqrt {I}}={\mathfrak {p}}}$, and such an I is ${\displaystyle {\mathfrak {p}}}$-primary if ${\displaystyle {\mathfrak {p}}}$ is maximal. This gives a local criterion for a minimal prime: a prime ideal ${\displaystyle {\mathfrak {p}}}$ is a minimal prime over I if and only if ${\displaystyle IR_{\mathfrak {p}}}$ is a ${\displaystyle {\mathfrak {p}}R_{\mathfrak {p}}}$-primary ideal. When R is a Noetherian ring, ${\displaystyle {\mathfrak {p}}}$ is a minimal prime over I if and only if ${\displaystyle R_{\mathfrak {p}}/IR_{\mathfrak {p}}}$ is an Artinian ring (i.e., ${\displaystyle {\mathfrak {p}}R_{\mathfrak {p}}}$ is nilpotent module I). The pre-image of ${\displaystyle IR_{\mathfrak {p}}}$ under ${\displaystyle R\to R_{\mathfrak {p}}}$ is a primary ideal of ${\displaystyle R}$ called the ${\displaystyle {\mathfrak {p}}}$-primary component of I.

For a minimal prime ideal ${\displaystyle {\mathfrak {p}}}$ in a local ring ${\displaystyle A}$, in general, it need not be the case that ${\displaystyle \dim A/{\mathfrak {p}}=\dim A}$, the Krull dimension of ${\displaystyle A}$.

A Noetherian local ring ${\displaystyle A}$ is said to be equidimensional if for each minimal prime ideal ${\displaystyle {\mathfrak {p}}}$, ${\displaystyle \dim A/{\mathfrak {p}}=\dim A}$. For example, a local Noetherian integral domain and a local Cohen–Macaulay ring are equidimensional.

See also equidimensional scheme and quasi-unmixed ring.

• Extension and contraction of ideals
• Normalization

• Eisenbud, David (1995), Commutative algebra, Graduate Texts in Mathematics, 150, Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-5350-1, ISBN 978-0-387-94268-1, MR 1322960
• Kaplansky, Irving (1974), Commutative rings, University of Chicago Press, MR 0345945

• http://stacks.math.columbia.edu/tag/035E
• http://stacks.math.columbia.edu/tag/035P