# DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS

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: huanyinchen@yahoo.cn http://huanyinchens.blogbus.com

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 Bibtex @ { ieja266354, journal = {International Electronic Journal of Algebra}, issn = {1306-6048}, eissn = {1306-6048}, address = {1710 Sokak, No:41, Batikent/Ankara}, publisher = {Abdullah HARMANCI}, year = {2009}, volume = {6}, number = {6}, pages = {107 - 114}, title = {DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS}, key = {cite}, author = {Chen, Huanyin} } APA Chen, H. (2009). DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS . International Electronic Journal of Algebra , 6 (6) , 107-114 . MLA Chen, H. "DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS" . International Electronic Journal of Algebra 6 (2009 ): 107-114 < Chicago Chen, H. "DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS". International Electronic Journal of Algebra 6 (2009 ): 107-114 RIS TY - JOUR T1 - DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS AU - HuanyinChen Y1 - 2009 PY - 2009 N1 - DO - T2 - International Electronic Journal of Algebra JF - Journal JO - JOR SP - 107 EP - 114 VL - 6 IS - 6 SN - 1306-6048-1306-6048 M3 - UR - Y2 - 2022 ER - EndNote %0 International Electronic Journal of Algebra DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS %A Huanyin Chen %T DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS %D 2009 %J International Electronic Journal of Algebra %P 1306-6048-1306-6048 %V 6 %N 6 %R %U ISNAD Chen, Huanyin . "DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS". International Electronic Journal of Algebra 6 / 6 (Aralık 2009): 107-114 . AMA Chen H. DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS. IEJA. 2009; 6(6): 107-114. Vancouver Chen H. DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS. International Electronic Journal of Algebra. 2009; 6(6): 107-114. IEEE H. Chen , "DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS", , c. 6, sayı. 6, ss. 107-114, Ara. 2009
• ISSN: 1306-6048
• Yayın Aralığı: Yılda Aylık
• Yayıncı: Abdullah HARMANCI

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