Abstract
A matrix A ∈ M n (R) is e-clean provided there exists an idempotent E ∈ M n (R) such that A-E ∈ 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 × n matrix over a Dedekind domain coincide for all n ⩾ 3. The analogous for (s, 2) property is also obtained.
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Chen, H. Clean matrices over commutative rings. Czech Math J 59, 145–158 (2009). https://doi.org/10.1007/s10587-009-0010-x
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DOI: https://doi.org/10.1007/s10587-009-0010-x