Matrix invertible extensions over commutative rings. Part II: determinant liftability

TL;DR

This paper investigates conditions under which 2x2 matrices over commutative rings can be extended or lifted to invertible matrices, providing criteria and applications related to determinant liftability and elementary divisor domains.

Contribution

It establishes necessary and sufficient conditions for determinant liftability of matrices over certain rings and connects these conditions to properties like elementary divisor domains.

Findings

Matrices over $\

determinant liftability is characterized for specific ring classes.

Connections are made between extendability and determinant liftability in ring theory.

Abstract

A unimodular $2\times 2$ matrix $A$ with entries in a commutative ring $R$ is called weakly determinant liftable if there exists a matrix $B$ congruent to $A$ modulo $R\det(A)$ and $\det(B)=0$; if we can choose $B$ to be unimodular, then $A$ is called determinant liftable. If $A$ is extendable to an invertible $3\times 3$ matrix $A^+$, then $A$ is weakly determinant liftable. If $A$ is simple extendable (i.e., we can choose $A^+$ such that its $(3,3)$ entry is $0$), then $A$ is determinant liftable. We present necessary and/or sufficient criteria for $A$ to be (weakly) determinant liftable and we use them to show that if $R$ is a $\Pi_2$ ring in the sense of Part I (resp.\ is a pre-Schreier domain), then $A$ is simply extendable (resp.\ extendable) iff it is determinant liftable (resp.\ weakly determinant liftable). As an application we show that each $J_{2,1}$ domain (as defined by Lorenzini) is an elementary divisor domain.