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

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 as(R)≤ 2, then R is an E₂ ring iff it is an SE₂ ring.