On the space of Sylvester matrix rank functions

Abstract

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 $\mathcal{D}$.