Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

WEAKLY STABLE CONDITIONS FOR EXCHANGE RINGS

  • Chen, Huanyin (DEPARTMENT OF MATHEMATICS HUNAN NORMAL UNIVERSITY)
  • Published : 2007.07.30
Download PDF
⟨ Previous Next ⟩

Abstract

A ring R has weakly stable range one provided that aR+bR=R implies that there exists a $y{\in}R$ such that $a+by{\in}R$ is right or left invertible. We prove, in this paper, that every regular element in an exchange ring having weakly stable range one is the sum of an idempotent and a weak unit. This generalize the corresponding result of one-sided unit-regular ring. Extensions of power comparability and power cancellation are also studied.

Keywords

References

  1. V. P. Camillo and D. Khurana, A characterization of unit regular rings, Comm. Algebra 29 (2001), no. 5, 2293-2295 https://doi.org/10.1081/AGB-100002185
  2. H. Chen, Elements in one-sided unit regular rings, Comm. Algebra 25 (1997), no. 8, 2517-2529 https://doi.org/10.1080/00927879708826002
  3. H. Chen, Comparability of modules over regular rings, Comm. Algebra 25 (1997), no. 11, 3531-3543 https://doi.org/10.1080/00927879708826068
  4. H. Chen, Exchange rings with artinian primitive factors, Algebr. Represent Theory 2 (1999), no. 2, 201-207 https://doi.org/10.1023/A:1009927211591
  5. H. Chen, Power-substitution, exchange rings and unit ${\pi}-regularity$, Comm. Algebra 28 (2000), no. 11, 5223-5233 https://doi.org/10.1080/00927870008827151
  6. H. Chen, Exchange rings satisfying power ideal-cancellation, Sci. China Ser. A 46 (2003), no. 6, 804-814 https://doi.org/10.1360/02ys0307
  7. K. R. Goodearl, von Neumann Regular Rings, Pitman, London, San Francisco, Melbourne, 1979; second ed., Krieger, Malabar, Fl., 1991
  8. Q. Li and W. Tong, Weak cancellation of modules and weakly stable range conditions for exchange rings, Acta Math. Sinica (Chinese) 45 (2002), no. 6, 1121-1126
  9. Q. Li, J. Zhu, and W. Tong, On related power comparability of modules, Comm. Algebra 31 (2003), no. 10, 4925-4938 https://doi.org/10.1081/AGB-120023140
  10. A. Tuganbaev, Rings Close to Regular, Kluwer Academic Publishers, Dordrecht, Boston, London, 2002
  11. J. Wei, Exchange rings with weakly stable range one, Vietnam J. Math. 32 (2004), no. 4, 441-449
  12. J. Wei, Unit-regularity and stable range conditions, Comm. Algebra 33 (2005), no. 6, 1937-1946 https://doi.org/10.1081/AGB-200063337
  13. T. Wu, The problem of weak cancellation of modules and qu-regular rings, Acta Math. Sinica (Chinese), 38 (1995), no. 6, 746-751
  14. T. Wu, The power-substitution condition of endomorphism rings of quasi-projective modules, Comm. Algebra 28 (2000), no. 1, 407-418 https://doi.org/10.1080/00927870008841081