= Krull dimension
{wiki}
The <Krull dimension> of a <commutative ring> $R$ is the <supremum> of the lengths $n$ of strict chains of <prime ideals> $\mathfrak p_0\subsetneq\cdots\subsetneq\mathfrak p_n$. The length counts inclusions, not the number of <ideals>.
A <field> has dimension zero. The <integers> have dimension one, with chains $(0)\subsetneq(p)$, and $k[X_1,\ldots,X_n]$ over a <field> has dimension $n$. In <algebraic geometry>, chains of primes correspond to nested irreducible closed subsets, so this algebraic definition captures geometric dimension. It is distinct from the <dimension of a vector space>.
Back to article page
