Krull dimension

Summary

The Krull dimension of a commutative ring is the largest length of a strictly increasing chain of prime ideals. It is the algebraic notion of dimension: the coordinate ring of an affine variety has Krull dimension equal to the dimension of the variety. Fields have dimension $0$, and principal ideal domains that are not fields, such as $\Z$, have dimension $1$. The polynomial ring in $n$ variables over a field has dimension $n$.