# 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.

## Definition

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 ; 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 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 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 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: 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*.

## 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.

See also equidimensional scheme and quasi-unmixed ring.

## Notes

- Kaplansky 1974, p. 6
- Kaplansky 1974, p. 59
- Eisenbud 1995, p. 47
- Kaplansky 1974, p. 16
- Kaplansky 1974, p. 57

## References

- 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