Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

ELEMENTARY MATRIX REDUCTION OVER ZABAVSKY RINGS

  • Chen, Huanyin (Department of Mathematics Hangzhou Normal University) ;
  • Sheibani, Marjan (Faculty of Mathematics Statistics and Computer Science Semnan University)
  • Received : 2015.01.21
  • Published : 2016.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

We prove, in this note, that a Zabavsky ring R is an elementary divisor ring if and only if R is a $B{\acute{e}}zout$ ring. Many known results are thereby generalized to much wider class of rings, e.g. [4, Theorem 14], [7, Theorem 4], [9, Theorem 1.2.14], [11, Theorem 4] and [12, Theorem 7].

Keywords

References

  1. D. D. Anderson and V. P. Camillo, Commutative rings whose elements are a sum of a unit and idempotent, Comm. Algebra 30 (2002), no. 7, 3327-3336. https://doi.org/10.1081/AGB-120004490
  2. H. Chen, Rings Related Stable Range Conditions, Series in Algebra 11, World Scientific, Hackensack, NJ, 2011.
  3. H. Chen, H. Kose, and Y. Kurtulmaz, On feckly clean rings, J. Algebra Appl. 14 (2015), no. 4, 1550046, 15 pp.
  4. O. V. Domsha and I. S. Vasiunyk, Combining local and adequate rings, Book of abstracts of the International Algebraic Conference, Taras Shevchenko National University of Kyiv, Kyiv, Ukraine, 2014.
  5. M. Henriksen, On a class of regular rings that are elementary divisor rings, Arch. Math. 24 (1973), 133-141. https://doi.org/10.1007/BF01228189
  6. W. W. McGovern, Bezout rings with almost stable range 1, J. Pure Appl. Algebra 212 (2008), no. 2, 340-348. https://doi.org/10.1016/j.jpaa.2007.05.026
  7. O. Pihura, Commutative Bezout rings in which 0 is adequate is a semi regular, preprint, 2015.
  8. M. Roitman, The Kaplansky condition and rings of almost stable range 1, Proc. Amer. Math. Soc. 141 (2013), no. 9, 3013-3018. https://doi.org/10.1090/S0002-9939-2013-11567-4
  9. B. V. Zabavsky, Diagonal Reduction of Matrices over Rings, Mathematical Studies Monograph Series, Vol. XVI, VNTL Publisher, 2012.
  10. B. V. Zabavsky, Questions related to the K-theoretical aspect of Bezout rings with various stable range conditions, Math. Stud. 42 (2014), 89-103.
  11. B. V. Zabavsky and S. I. Bilavska, Every zero adequate ring is an exchange ring, J. Math. Sci. 187 (2012), 153-156. https://doi.org/10.1007/s10958-012-1058-y
  12. B. V. Zabavsky and M. Y. Komarnyts'kyi, Cohn-type theorem for adequacy and elementary divisor rings, J. Math. Sci. 167 (2010), 107-111. https://doi.org/10.1007/s10958-010-9906-0