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

## Properties

All rings are assumed to be commutative and unital.