  Given a ring $R$, the notion of Sylvester rank function was conceived within
the context of Cohn's classification theory of epic division $R$-rings.
  In this paper we study and describe the space of Sylvester rank functions on
certain families of rings, including Dedekind domains, simple left noetherian
rings and skew Laurent polynomial rings $\mathcal{D}[t^{\pm 1};\tau]$ for any
division ring $\mathcal{D}$ and any automorphism $\tau$ of $\mathcal{D}$.
