Definition:Height (Ring Theory)

Definition

Let $A$ be a commutative ring.

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:

$\mathfrak p_0 \subsetneqq \mathfrak p_1 \subsetneqq \cdots \subsetneqq \mathfrak p$