Codimension of an ideal

Definition

Let $R$ be a commutative unital ring and $I$ an ideal in $R$. The codimension or height of $I$ is defined as follows:

• If $I$ is a prime ideal, it is defined as the Krull dimension of the localization $R_{I}$
• Otherwise, it is defined to be the minimum of the Krull dimensions of prime ideals containing $I$