Krull dimension in modal logic

We develop the theory of Krull dimension for S4-algebras and Heyting algebras. This leads to the concept of modal Krull dimension for topological spaces. We compare modal Krull dimension to other well-known dimension functions, and show that it can detect differences between topological spaces that Krull dimension is unable to detect. We prove that for a T₁-space to have a finite modal Krull dimension can be described by an appropriate generalization of the well-known concept of a nodec space. This, in turn, can be described by modal formulas zem_n which generalize the well-known Zeman formula zem. We show that the modal logic S4.Z_n:= S4 + zem_n is the basic modal logic of T₁-spaces of modal Krull dimension ≤ n, and we construct a countable dense-in-itself ω-resolvable Tychonoff space Z_n of modal Krull dimension n such that S4.Z_n is complete with respect to Z_n. This yields a version of the McKinsey-Tarski theorem for S4.Z_n. We also show that no logic in the interval [S4_n+1S4.Z_n) is complete with respect to any class of T₁-spaces.