# 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 Hauptidealsatz 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 that we do not exclude *I* even if it is a prime ideal.) 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 ring *R* is precisely a minimal associated prime (also called isolated prime) of .

## 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 power of*p*), then*p*is the unique minimal prime ideal over*I*.

## Properties

All rings are assumed to be unital.

- Every proper ideal
*I*in a commutative ring has at least one minimal prime ideal above it. The proof of this fact uses Zorn's lemma Template:Harv. 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 commutative Noetherian ring, there are only finitely many minimal prime ideals over any given ideal. Template:Harv, Template:Harv.
- The radical of any proper ideal
*I*coincides with the intersection of the minimal prime ideals over*I*. Template:Harv. - The set of zero divisors of a given commutative ring contains the union of the minimal prime ideals Template:Harv.
- Krull's Hauptidealsatz describes important properties of minimal prime ideals.

## References

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