Matrix invertible extensions over commutative rings. Part I: General theory

Abstract

A unimodular 2×2 matrix with entries in a commutative R is called extendable (resp. simply extendable) if it extends to an invertible 3×3 matrix (resp. invertible 3×3 matrix whose (3,3) entry is 0). We obtain necessary and sufficient conditions for a unimodular 2×2 matrix to be extendable (resp. simply extendable) and use them to study the class E2 (resp. SE2) of rings R with the property that all unimodular 2×2 matrices with entries in R are extendable (resp. simply extendable). We also study the larger class Π2 of rings R with the property that all unimodular 2×2 matrices of determinant 0 and with entries in R are (simply) extendable (e.g., rings with trivial Picard groups or pre-Schreier domains). Among Dedekind domains, polynomial rings over Z and Hermite rings, only the EDRs belong to the class E2 or SE2. If R has stable range at most 2 (e.g., R is a Hermite ring or dim⁡(R)≤1), then R is an E2 ring iff it is an SE2 ring.