Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BOUNDED MATRICES OVER REGULAR RINGS

  • Wang Shuqin (DEPARTMENT OF MATHEMATICS, SHANDONG ECONOMIC UNIVERSITY) ;
  • Chen Huanyin (DEPARTMENT OF MATHEMATICS, ZHEJIANG NORMAL UNIVERSITY)
  • Published : 2006.02.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we investigate bounded matrices over regular rings. We observe that every bounded matrix over a regular ring can be described by idempotent matrices and invertible matrices. Let A, $B{in}M_n(R)$ be bounded matrices over a regular ring R. We prove that $(AB)^d = U(BA)^dU^{-1}$ for some $U{\in}GL_n(R)$.

Keywords

References

  1. P. Ara, Strongly $\pi$-regular rings have stable range one, Proc. Amer. Math. Soc. 124 (1996), no. 11, 3293-3298
  2. P. Ara, K. R. Goodearl, K. C. O'Meara, and E. Pardo, Diagonalization of ma- trices over regular rings, Linear Algebra Appl. 265 (1997), 147-163 https://doi.org/10.1016/S0024-3795(96)00596-4
  3. M. J. Canfell, Completion of diagrams by automorphisms and Bass' first stable range condition, J. Algebra 176 (1995), no. 2, 480-503 https://doi.org/10.1006/jabr.1995.1255
  4. J. J. Carmona, J. Cuf and P. Menal, On the unit 1-stable rank of rings of analytic functions, Publ. Mat. 36 (1992), no. 2A, 439-447 https://doi.org/10.5565/PUBLMAT_362A92_08
  5. H. Chen, Exchange rings with Artinian primitive factors, Algebr. Represent. Theory 2 (1999), no. 2, 201-207 https://doi.org/10.1023/A:1009927211591
  6. H. Chen, Units, idempotents, and stable range conditions, Comm. Algebra 29 (2001), no. 2, 703-717 https://doi.org/10.1081/AGB-100001535
  7. H. Chen, Exchange rings having stable range one, Int. J. Math. Math. Sci. 25 (2001), no. 12, 763-770 https://doi.org/10.1155/S016117120100552X
  8. H. Chen, Extensions of rings with many units, Comm. Algebra 31 (2003), no. 5, 2403-2416 https://doi.org/10.1081/AGB-120019003
  9. K. R. Goodearl, von Neumann regular rings, Pitman, London, San Francisco, Melbourne, 1979
  10. K. R. Goodearl, von Neumann regular rings, second ed., Krieger, Malabar, Fl., 1991
  11. M. Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math.(Basel) 24 (1973), 133-141 https://doi.org/10.1007/BF01228189
  12. P. Menal and J. Moncasi, Lifting units in self-injective rings and an index theory for Rickart C$^{\ast}$-algebras, Pacific J. Math. 126 (1987), no. 2, 295-329 https://doi.org/10.2140/pjm.1987.126.295
  13. A. A. Klein, The sum of nil one-sided ideals of bounded index of a ring, Israel J. Math. 88 (1994), no. 1-3, 25-30 https://doi.org/10.1007/BF02937505
  14. K. R. Nagarajan, M. P. Devasahayam and T. Soundararajan, Products of three triangular matrices, Linear Algeb. Appl. 292 (1999), no. 1-3, 61-71 https://doi.org/10.1016/S0024-3795(99)00003-8
  15. K. R. Nagarajan, M. P. Devasahayam and T. Soundararajan, Products of three triangular matrices over commutative rings, Linear Algebra Appl. 348 (2002), 1-6 https://doi.org/10.1016/S0024-3795(01)00453-0
  16. W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592 https://doi.org/10.1080/00927879908826649
  17. L. N. Vaseretein and E. Wheland, Commutators and companion matrices over rings of stable rank 1, Linear Algebra Appl. 142 (1990), 263-277 https://doi.org/10.1016/0024-3795(90)90270-M