DIAGONALIZATION OF REGULAR MATRICES OVER EXCHANGE RINGS

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.

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  • P. Ara, K.R. Goodearl, K.C. O’Meara and E. Pardo, Diagonalization of ma- trices over regular rings, Linear Algebra Appl., 265 (1997), 146–163.
  • K.I. Beidar, K.C. O’Meara and R.M. Raphael, On uniform diagonalisation of matrices over regular rings and one-accessible regular algebras, Comm. Alge- bra, 32 (2004), 3543–3562.
  • H. Chen, Regular rings with finite stable range, Comm. Algebra, 29 (2001), –166.
  • H. Chen, Exchange rings over which every regular matrix admits diagonal re- duction, J. Algebra Appl., 3 (2004), 207–217.
  • H. Chen, Diagonal reductions of matrices over exchange ideals, Czechoslovak Math. Math., 56 (2006), 9–18.
  • H. Chen, On exchange hermitian rings, Algebra Colloq., to appear. D. Huylebrouck, Diagonal and von Neumann regular matrices over a Dedekind domain, Portugal. Math., 51 (1994), 291–303.
  • K.C. O’Meara and R. M. Raphael, Uniform diagonalisation of matrices over regular rings, Algebra Univers., 45 (2001), 383–405.
  • K.C. O’Meara and C. Vinsonhaler, On approximately simultaneously diagonal- izable matrices, Linear Algebra Appl., 412 (2006), 39–74.
  • G. Song and X. Guo, Diagonability of idempotent matrices over noncommuta- tive rings, Linear Algebra Appl., 297 (1999), 1–7.
  • A.A. Tuganbaev, Rings Close to Regular, Kluwer Academic Publishers, Dor- drecht, Boston, London, 2002.
  • B.V. Zabavsky, Reduction of matrices over B´ezout rings of stable rank not high than 2, Ukrainian Math. J., 55 (2003), 665–670. Huanyin Chen
  • 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