  This paper is the English translation of the first 4 sections of the article
"Thierry Coquand, Henri Lombardi, and Claude Quitt\'e. Dimension de Heitmann
des treillis distributifs et des anneaux commutatifs. Publications
Math\'ematiques de l'Universit\'e de Franche-Comt\'e, Besan\c{c}on. Alg\`ebre
et th\'eorie des nombres. (2006)", after some corrections.
  Sections 5-7 of the original article are treated a bit more simply in the
book "Henri Lombardi and Claude Quitt\'e. Commutative algebra: constructive
methods. Finite projective modules. Algebra and applications, 20, Springer,
Dordrecht, 2015."
  We study the notion of dimension introduced by Heitmann in his remarkable
article "Raymond Heitmann. Generating non-Noetherian modules efficiently, Mich.
Math. J., 31, (1084)" as well as a related notion, only implicit in his proofs.
We first develop this within the general framework of the theory of
distributive lattices and spectral spaces.
  Keywords: Constructive mathematics, distributive lattice, Heyting algebra,
spectral space, Zariski lattice, Zariski spectrum, Krull dimension, maximal
spectrum, Heitmann lattice, Heitmann spectrum, Heitmann dimensions.
  MSC: 13C15, 03F65, 13A15, 13E05
