  For a unital ring R, a Sylvester rank function is a numerical invariant which
can be described in 3 equivalent ways: on finitely presented left R-modules, or
on rectangular matrices over R, or on maps between finitely generated
projective left R-modules. We extend each Sylvester rank function to all pairs
of left R-modules $M_1\subseteq M_2$, and to all maps between left R-modules
satisfying suitable properties including continuity and additivity.
  As an application, we show that for any epimorphism $R\rightarrow S$ of
unital rings, the pull-back map from the set of Sylvester rank functions of S
to that of R is injective. We also give a new proof of Schofield's result
describing the image of this map when S is the universal localization of R
inverting a set of maps between finitely generated projective left R-modules.
