# Definition:Height (Ring Theory)

Jump to navigation
Jump to search

## 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:

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