Definition:Height of Prime Ideal

Definition
Let $A$ be a commutative ring with unity.

Let $\mathfrak p$ be a prime ideal in $A$.

The height of $\mathfrak p$ is the supremum over all $n$ such that there exists a chain of prime ideals:


 * $\mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p_n = \mathfrak p$

It is denoted by $\map {\mathrm {ht} } {\mathfrak p}$.