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.\ $SE_2$) 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 $\mathbb Z$ and Hermite rings, only the EDRs belong to the class $E_2$ or $SE_2$. If $as(R)\le 2$, then $R$ is an $E_2$ ring iff it is an $SE_2$ ring.