height of a prime ideal

Let $R$ be a commutative ring and $\\mathfrak{p}$ a prime ideal of $R$ . The height of $\\mathfrak{p}$ is the supremum of all integers $n$ such that there exists a chain

\\mathfrak{p}_{0}\\subset\\cdots\\subset\\mathfrak{p}_{n}=\\mathfrak{p}

of distinct prime ideals. The height of $\\mathfrak{p}$ is denoted by $\\operatorname{h}(\\mathfrak{p})$ .

$\\operatorname{h}(\\mathfrak{p})$ is also known as the rank of $\\mathfrak{p}$ and the codimension of $\\mathfrak{p}$ .

The Krull dimension of $R$ is the supremum of the heights of all the prime ideals of $R$ :

\\sup\\{\\operatorname{h}(\\mathfrak{p})\\mid\\mathfrak{p}\\mbox{ prime in }R\\}.