DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS

Year 2009, Volume: 6 Issue: 6, 107 - 114, 01.12.2009

Huanyin Chen

Abstract

In this paper, we establish several necessary and sufficient conditions under which every regular matrix admits a diagonal reduction. We prove that every regular matrix over an exchange ring R admits diagonal reduction if and only if for any m, n ∈ N (m ≥ n + 1) and any regular X ∈ Mm×n(R), ³X 0m×(m−n)´∈ Mm(R) is unit-regular if and only if for any m, n ∈ N (m ≥ n+ 1) and any regular X ∈ Mm×n(R), there exist an idempotent E ∈ Mm(R) and a completed U ∈ Mm×n(R) such that X = EU if and only if for any idempotents e ∈ R, f ∈ M2(R), ϕ : eR ∼= f(2R) implies that there exists a completed u ∈ 2R such that ϕ(e) = ue = fu. These shows that diagonal reduction over exchange rings behaves like stable ranges.

Keywords

diagonal reduction, exchange ring, unit-regularity

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• Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, People’s Republic of China e-mail: [e-mail] http://huanyinchens.blogbus.com