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.
Keywords:
diagonal reduction, exchange ring unit-regularity,
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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
- ISSN: 1306-6048
- Yayın Aralığı: Yılda 2 Sayı
- Başlangıç: 2007
- Yayıncı: Abdullah HARMANCI