On the space of Sylvester matrix rank functions

TL;DR

This paper explores the structure and properties of Sylvester matrix rank functions across various classes of rings, including Dedekind domains and skew Laurent polynomial rings, extending the understanding of their classification.

Contribution

It characterizes the space of Sylvester rank functions for specific rings such as Dedekind domains, simple noetherian rings, and skew Laurent polynomial rings, providing new insights into their structure.

Findings

Describes the space of Sylvester rank functions for Dedekind domains.

Analyzes Sylvester rank functions on simple left noetherian rings.

Studies Sylvester rank functions on skew Laurent polynomial rings.

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 $\mathcal{D}$.