Clean matrices over commutative rings

A matrix A a M (n) (R) is e-clean provided there exists an idempotent E a M (n) (R) such that A-E a GL (n) (R) and det E = e. We get a general criterion of e-cleanness for the matrix [[a (1), a (2),..., a (n) +1]]. Under the n-stable range ondition, it is shown that [[a (1), a (2),..., a (n) +1]] is 0-clean iff (a (1), a (2),..., a (n) +1) = 1. As an application, we prove that the 0-cleanness and unit-regularity for such n x n matrix over a Dedekind domain coincide for all n a (c) 3/4 3. The analogous for (s, 2) property is also obtained.