p0 ⊂ p1
⊂ … ⊂ pn ⊂ R
but there is no longer such chain.
If X ⊆ Kn is a subset of the nth Cartesian power of an algebraically closed field defined by the vanishing of finitely many polynomials f1(x1, … xn), … fm(x1, … xn) ∈ K[x1, …, xn], then the Krull dimension of X is defined to be the Krull dimension of the ring K[x1, … xn]/(f1, … fm).
If X ⊆ Kn is any definable set in an algebraically closed field, then X may be expressed as a finite union of sets of the form Y - Z where Y and Z are defined by polynomial equations and the Krull dimension of Z is less than that of Y. We define the Krull dimension of X to be the maximal Krull dimension of such a Y.