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 formula_1; this follows for instance from the primary decomposition of "I".
Properties.
All rings are assumed to be commutative and unital.
Equidimensional ring.
For a minimal prime ideal formula_27 in a local ring formula_42, in general, it need not be the case that formula_43, the Krull dimension of formula_42.
A Noetherian local ring formula_42 is said to be equidimensional if for each minimal prime ideal formula_27, formula_43. For example, a local Noetherian integral domain and a local Cohen–Macaulay ring are equidimensional.
See also equidimensional scheme and quasi-unmixed ring.