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This paper is the English translation of the first 4 sections of the article"Thierry Coquand, Henri Lombardi, and Claude Quitt\'e. Dimension de Heitmanndes treillis distributifs et des anneaux commutatifs. PublicationsMath\'ematiques de l'Universit\'e de Franche-Comt\'e, Besan\c{c}on. Alg\`ebreet th\'eorie des nombres. (2006)", after some corrections.  Sections 5-7 of the original article are treated a bit more simply in thebook "Henri Lombardi and Claude Quitt\'e. Commutative algebra: constructivemethods. Finite projective modules. Algebra and applications, 20, Springer,Dordrecht, 2015."  We study the notion of dimension introduced by Heitmann in his remarkablearticle "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 ofdistributive lattices and spectral spaces.  Keywords: Constructive mathematics, distributive lattice, Heyting algebra,spectral space, Zariski lattice, Zariski spectrum, Krull dimension, maximalspectrum, Heitmann lattice, Heitmann spectrum, Heitmann dimensions.  MSC: 13C15, 03F65, 13A15, 13E05

Heitman dimension of distributive lattices and commutative rings