DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS

Authors : Huanyin CHEN

Pages : 107-114

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Publication Date : 2009-12-01

Article Type : Research Paper

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 insert ignore into journalissuearticles values(m ≥ n + 1); and any regular X ∈ Mm×ninsert ignore into journalissuearticles values(R);, ³X 0m×insert ignore into journalissuearticles values(m−n);´∈ Mminsert ignore into journalissuearticles values(R); is unit-regular if and only if for any m, n ∈ N insert ignore into journalissuearticles values(m ≥ n+ 1); and any regular X ∈ Mm×ninsert ignore into journalissuearticles values(R);, there exist an idempotent E ∈ Mminsert ignore into journalissuearticles values(R); and a completed U ∈ Mm×ninsert ignore into journalissuearticles values(R); such that X = EU if and only if for any idempotents e ∈ R, f ∈ M2insert ignore into journalissuearticles values(R);, ϕ : eR ∼= finsert ignore into journalissuearticles values(2R); implies that there exists a completed u ∈ 2R such that ϕinsert ignore into journalissuearticles values(e); = ue = fu. These shows that diagonal reduction over exchange rings behaves like stable ranges.
Keywords : diagonal reduction, exchange ring, unit regularity