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

A unimodular $2\times 2$ matrix with entries in a commutative R is called extendable (resp. simply extendable) if it extends to an invertible $3\times 3$ matrix (resp. invertible $3\times 3$ matrix whose $(3,3)$ entry is 0). We obtain necessary and sufficient conditions for a unimodular $2\times 2$ matrix to be extendable (resp. simply extendable) and use them to study the class $E_2$ (resp. $S{E}_2$) of rings R with the property that all unimodular $2\times 2$ matrices with entries in R are extendable (resp. simply extendable). We also study the larger class ${\Pi }_2$ of rings R with the property that all unimodular $2\times 2$ 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 $E_2$ or $S{E}_2$. If R has stable range at most 2 (e.g., R is a Hermite ring or $\dim \left(R\right)\le 1$), then R is an $E_2$ ring iff it is an $S{E}_2$ ring.