DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS

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.