In mathematics, especially in 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 prime ideals.

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 R/I; this follows for instance from the primary decomposition of I.

Examples

Properties

All rings are assumed to be commutative and unital.

Equidimensional ring

For a minimal prime ideal in a local ring A, in general, it need not be the case that, the Krull dimension of A. A Noetherian local ring A 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.