  We present constructive versions of Krull's dimension theory for commutative
rings and distributive lattices. The foundations of these constructive versions
are due to Joyal, Espa\~nol and the authors. We show that the notion of Krull
dimension has an explicit computational content in the form of existence (or
lack of existence) of some algebraic identities. We can then get an explicit
computational content where abstract results about dimensions are used to show
the existence of concrete elements. This can be seen as a partial realisation
of Hilbert's program for classical abstract commutative algebra.
