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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 ₂ 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.
Cǎlugǎreanu et al. (Mon,) studied this question.