  Inspired by the construction of the Cuntz semigroup for a C*-algebra, we
introduce the matrix Malcolmson semigroup and the finitely presented module
Malcolmson semigroup for a unital ring. These two semigroups are shown to have
isomorphic Grothendieck group in general and be isomorphic for von Neumann
regular rings. For unital C*-algebras, it is shown that the matrix Malcolmson
semigroup has a natural surjective order-preserving homomorphism to the Cuntz
semigroup, every dimension function is a Sylvester matrix rank function, and
there exist Sylvester matrix rank functions which are not dimension functions.
