  We introduce the class E2 (resp. SE2) of commutative rings R with the
property that each unimodular 2 x 2 matrix with entries in R extends to an
invertible 3 x 3 matrix (resp. invertible 3 x 3 matrix whose (3, 3) entry is
0). Among noetherian domains of dimension 1, polynomial rings over Z or Hermite
rings, only EDRs belong to the class. Using this, stable ranges, and units and
projective modules interpretations, we reobtain (often refine) criteria due to
Kaplansky, McGovern, Roitman, Shchedryk, Wiegand and obtain new criteria for a
Hermite ring to be an EDR. For instance, we characterize Hermite rings which
are EDRs by (1) means of equations involving unimodular triples and (2)
surjectivity of some maps involving units of factor rings by principal ideals.
We use these criteria to show that each Bezout ring R that is either a J2,1
domain or an (SU)2 ring (as introduced by Lorenzini) such that for each nonzero
a 2 R there exists no nontrivial self-dual projective R/Ra-module of rank 1
generated by 2 elements (e.g., all its elements are squares), is an EDR, thus
solving or partially solving problems raised by Lorenzini. Many other classes
of rings are introduced and studied. To ease the reading, this version
concentrates only on extendability properties of unimodular 2 x 2 matrices and
hence does not consider the case n > 2.
