Clean matrices over commutative rings

A matrix A ∈ Mn(R) is e-clean provided there exists an idempotent E ∈ Mn(R) such that A-E ∈ GLn(R) and det E = e. We get a general criterion of e-cleanness for the matrix [[a1, a2,..., an+1]]. Under the n-stable range ondition, it is shown that [[a1, a2,..., an+1]] is 0-clean iff (a1, a2,..., an+1) = 1. As an application, we prove that the 0-cleanness and unit-regularity for such n × n matrix over a Dedekind domain coincide for all n ⩾ 3. The analogous for (s, 2) property is also obtained.