# Krull dimension

If $R$ is a ring, the Krull dimension (or simply dimension) of $R$, $\dim R$ is the supremum of all integers $n$ such that there is an increasing sequence of prime ideals $\mathfrak{p}_{0}\subsetneq\cdots\subsetneq\mathfrak{p}_{n}$ of length $n$ in $R$.

If $X$ is a topological space, the Krull dimension (or simply dimension) of $X$, $\dim X$ is the supremum of all integers $n$ such that there is a decreasing sequence of irreducible closed subsets $F_{0}\supsetneq\cdots\supsetneq F_{n}$ of $X$.

Title Krull dimension KrullDimension 2013-03-22 12:03:27 2013-03-22 12:03:27 mathcam (2727) mathcam (2727) 8 mathcam (2727) Definition msc 54-00 dimension (Krull) HeightOfAPrimeIdeal Dimension3