Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

QUASIPOLAR MATRIX RINGS OVER LOCAL RINGS

  • Cui, Jian (Department of Mathematics Anhui Normal University) ;
  • Yin, Xiaobin (Department of Mathematics Anhui Normal University)
  • Received : 2013.04.22
  • Published : 2014.05.31
Download PDF
⟨ Previous Next ⟩

Abstract

A ring R is called quasipolar if for every a 2 R there exists $p^2=p{\in}R$ such that $p{\in}comm^2{_R}(a)$, $ a+p{\in}U(R)$ and $ap{\in}R^{qnil}$. The class of quasipolar rings lies properly between the class of strongly ${\pi}$-regular rings and the class of strongly clean rings. In this paper, we determine when a $2{\times}2$ matrix over a local ring is quasipolar. Necessary and sufficient conditions for a $2{\times}2$ matrix ring to be quasipolar are obtained.

Keywords

References

  1. G. Borooah, A. J. Diesl, and T. J. Dorsey, Strongly clean triangular matrix rings over local rings, J. Algebra 312 (2007), no. 2, 773-797. https://doi.org/10.1016/j.jalgebra.2006.10.029
  2. G. Borooah, A. J. Diesl, and T. J. Dorsey, Strongly clean matrix rings over commutative local rings, J. Pure Appl. Algebra 212 (2008), no. 1, 281-296. https://doi.org/10.1016/j.jpaa.2007.05.020
  3. J. Chen, X. Yang, and Y. Zhou, When is the $2{\times}2$ matrix ring over a commutative local ring strongly clean?, J. Algebra 301 (2006), no. 1, 280-293. https://doi.org/10.1016/j.jalgebra.2005.08.005
  4. J. Chen, X. Yang, and Y. Zhou, On strongly clean matrix and triangular matrix rings, Comm. Algebra 34 (2006), no. 10, 3659-3674. https://doi.org/10.1080/00927870600860791
  5. P. M. Cohn, Free Rings and Their Relations, 2nd edn, Academic Press, 1985.
  6. J. Cui and J. Chen, When is a $2{\times}2$ matrix ring over a commutative local ring quasipo- lar?, Comm. Algebra 39 (2011), no. 9, 3212-3221. https://doi.org/10.1080/00927872.2010.499118
  7. J. Cui and J. Chen, Characterizations of quasipolar rings, Comm. Algebra 41 (2013), no. 9, 3207- 3217. https://doi.org/10.1080/00927872.2012.660670
  8. M. F. Dischinger, Sur les anneaux fortement ${\pi}$-reguliers, C. R. Math. Acad. Sci. Paris 283 (1976), no. 8, 571-573.
  9. J. Han and W. K. Nicholson, Extensions of clean rings, Comm. Algebra 29 (2001), no. 6, 2589-2595. https://doi.org/10.1081/AGB-100002409
  10. R. E. Harte, On quasinilpotents in rings, Panam. Math. J. 1 (1991), 10-16.
  11. J. J. Koliha and P. Patricio, Elements of rings with equal spectral idempotents, J. Aust. Math. Soc. 72 (2002), no. 1, 137-152. https://doi.org/10.1017/S1446788700003657
  12. B. Li, Strongly clean matrix rings over noncommutative local rings, Bull. korean Math. Soc. 46 (2009), no. 1, 71-78.
  13. Y. Li, Strongly clean matrix rings over local rings, J. Algebra 312 (2007), no. 1, 397-404. https://doi.org/10.1016/j.jalgebra.2006.10.032
  14. W. K. Nicholson, Local group rings, Canad. Math. Bull. 15 (1972), 137-138. https://doi.org/10.4153/CMB-1972-025-1
  15. W. K. Nicholson, Strongly clean rings and Fitting's lemma, Comm. Algebra 27 (1999), no. 8, 3583-3592. https://doi.org/10.1080/00927879908826649
  16. Z. Wang and J. Chen, On two open problems about strongly clean rings, Bull. Aust. Math. Soc. 70 (2004), no. 2, 279-282. https://doi.org/10.1017/S0004972700034493
  17. X. Yang and Y. Zhou, Strong cleanness of the $2{\times}2$ matrix ring over a general local ring, J. Algebra 320 (2008), no. 6, 2280-2290. https://doi.org/10.1016/j.jalgebra.2008.06.012
  18. Z. Ying and J. Chen, On quasipolar rings, Algebra Colloq. 19 (2012), no. 4, 683-692. https://doi.org/10.1142/S1005386712000557