Matrix invertible extensions over commutative rings. Part III: Hermite rings

TL;DR

This paper refines and introduces new algebraic criteria to determine when Hermite rings and Bézout domains are Euclidean division rings (EDRs), expanding understanding of their structure and properties.

Contribution

It provides new and refined criteria for Hermite rings to be EDRs, including conditions involving unit groups, matrix lifting, and unimodular pairs, with applications to specific ring classes.

Findings

Hermite rings are EDRs if certain unit group homomorphisms are surjective.

Reduced Hermite rings are EDRs if they are pre-Schreier and matrices lift appropriately.

Bézout domains are EDDs if specific unimodular pair conditions are met.

Abstract

We reobtain and often refine prior criteria due to Kaplansky, McGovern, Roitman, Shchedryk, Wiegand, and Zabavsky--Bilavska and obtain new criteria for a Hermite ring to be an \textsl{EDR}. We mention three criteria: (1) a Hermite ring $R$ is an \textsl{EDR} iff for all pairs $(a,c)\in R^2$, the product homomorphism $U(R/Rac)\times U\bigl(R/Rc(1-a)\bigr)\to U(R/Rc)$ between groups of units is surjective; (2) a reduced Hermite ring $R$ is an \textsl{EDR} iff it is a pre-Schreier ring and for each $a\in R$, every zero determinant unimodular $2\times 2$ matrix with entries in $R/Ra$ lifts to a zero determinant matrix with entries in $R$; (3) a B\'{e}zout domain $R$ is an \textsl{EDD} iff for all triples $(a,b,c)\in R^3$ there exists a unimodular pair $(e,f)\in R^2$ such that $(a,e)$ and $(be+af,1-a-bc)$ are unimodular pairs. We use these criteria to show that each B\'{e}zout ring $R$ that is an $(SU)_2$ ring (as introduced by Lorenzini) such that for each nonzero $a\in R$ there exists no nontrivial self-dual projective $R/Ra$-module of rank $1$ generated by $2$ elements (e.g., all its elements are squares), is an \textsl{EDR}.