Research Article
BibTex RIS Cite

$\mathcal{L}$-STABLE RINGS

Year 2021, , 63 - 94, 05.01.2021
https://doi.org/10.24330/ieja.852012

Abstract

If $\mathcal{L}(R)$ is a set of left ideals defined in
any ring $R,$ we say that $R$ is $\mathcal{L}$-stable if it has stable range
1 relative to the set $\mathcal{L}(R)$. We explore $\mathcal{L}$-stability
in general, characterize when it passes to related classes of rings, and
explore which classes of rings are $\mathcal{L}$-stable for some$\mathcal{\ L}.$ Some well known examples of $\mathcal{L}$-stable rings are presented,
and we show that the Dedekind finite rings are $\mathcal{L}$-stable for a
suitable $\mathcal{L}$.

References

  • D. D. Anderson, M. Axtell, S. J. Forman and J. Stickles, When are associates unit multiples?, Rocky Mountain J. Math., 34 (2004), 811-828.
  • H. Bass, K-Theory and stable algebra, Inst. Hautes tudes Sci. Publ. Math., 22 (1964), 5-60.
  • V. Camillo and H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra, 22 (1994), 4737-4749.
  • M. J. Canfell, Completion of diagrams by automorphisms and Bass' first stable range condition, J. Algebra, 176 (1995), 480-503.
  • H. Chen, On partially unit-regularity, Kyungpook Math. J., 42 (2002), 13-19.
  • H. Chen and W. K. Nicholson, Stable modules and a theorem of Camillo and Yu, J. Pure Appl. Algebra, 218 (2014), 1431-1442.
  • G. Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc., 216 (1976), 81-90.
  • D. Estes and J. Ohm, Stable range in commutative rings, J. Algebra, 7 (1967), 343-362.
  • K. R. Goodearl, Von Neumann Regular Rings, Second Edition, Krieger Publishing Co., Malabar, 1991.
  • H. K. Grover and D. Khurana, Some characterizations of VNL rings, Comm. Algebra, 37 (2009), 3288-3305.
  • I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
  • D. Khurana and T. Y. Lam, Rings with internal cancellation, J. Algebra, 284 (2005), 203-235.
  • T.Y. Lam, A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl., 3(3) (2004), 301-343.
  • W. K. Nicholson, Rings whose elements are quasi-regular or regular, Aequationes Math., 9 (1973), 64-70.
  • W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
  • W. K. Nicholson, On exchange rings, Comm. Algebra, 25 (1997), 1917-1918.
  • W. K. Nicholson, Annihilator-stability and unique generation, J. Pure Appl. Algebra, 221 (2017), 2557-2572.
  • W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
  • L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra, 34 (1984), 319-330.
  • R. B. Warfield, Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31-36.
Year 2021, , 63 - 94, 05.01.2021
https://doi.org/10.24330/ieja.852012

Abstract

References

  • D. D. Anderson, M. Axtell, S. J. Forman and J. Stickles, When are associates unit multiples?, Rocky Mountain J. Math., 34 (2004), 811-828.
  • H. Bass, K-Theory and stable algebra, Inst. Hautes tudes Sci. Publ. Math., 22 (1964), 5-60.
  • V. Camillo and H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra, 22 (1994), 4737-4749.
  • M. J. Canfell, Completion of diagrams by automorphisms and Bass' first stable range condition, J. Algebra, 176 (1995), 480-503.
  • H. Chen, On partially unit-regularity, Kyungpook Math. J., 42 (2002), 13-19.
  • H. Chen and W. K. Nicholson, Stable modules and a theorem of Camillo and Yu, J. Pure Appl. Algebra, 218 (2014), 1431-1442.
  • G. Ehrlich, Units and one-sided units in regular rings, Trans. Amer. Math. Soc., 216 (1976), 81-90.
  • D. Estes and J. Ohm, Stable range in commutative rings, J. Algebra, 7 (1967), 343-362.
  • K. R. Goodearl, Von Neumann Regular Rings, Second Edition, Krieger Publishing Co., Malabar, 1991.
  • H. K. Grover and D. Khurana, Some characterizations of VNL rings, Comm. Algebra, 37 (2009), 3288-3305.
  • I. Kaplansky, Elementary divisors and modules, Trans. Amer. Math. Soc., 66 (1949), 464-491.
  • D. Khurana and T. Y. Lam, Rings with internal cancellation, J. Algebra, 284 (2005), 203-235.
  • T.Y. Lam, A crash course on stable range, cancellation, substitution and exchange, J. Algebra Appl., 3(3) (2004), 301-343.
  • W. K. Nicholson, Rings whose elements are quasi-regular or regular, Aequationes Math., 9 (1973), 64-70.
  • W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
  • W. K. Nicholson, On exchange rings, Comm. Algebra, 25 (1997), 1917-1918.
  • W. K. Nicholson, Annihilator-stability and unique generation, J. Pure Appl. Algebra, 221 (2017), 2557-2572.
  • W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge University Press, Cambridge, 2003.
  • L. N. Vaserstein, Bass's first stable range condition, J. Pure Appl. Algebra, 34 (1984), 319-330.
  • R. B. Warfield, Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31-36.
There are 20 citations in total.

Details

Primary Language English
Subjects Mathematical Sciences
Journal Section Articles
Authors

Ayman M. A. Horoub This is me

W. K. Nıcholson This is me

Publication Date January 5, 2021
Published in Issue Year 2021

Cite

APA Horoub, A. M. A., & Nıcholson, W. K. (2021). $\mathcal{L}$-STABLE RINGS. International Electronic Journal of Algebra, 29(29), 63-94. https://doi.org/10.24330/ieja.852012
AMA Horoub AMA, Nıcholson WK. $\mathcal{L}$-STABLE RINGS. IEJA. January 2021;29(29):63-94. doi:10.24330/ieja.852012
Chicago Horoub, Ayman M. A., and W. K. Nıcholson. “$\mathcal{L}$-STABLE RINGS”. International Electronic Journal of Algebra 29, no. 29 (January 2021): 63-94. https://doi.org/10.24330/ieja.852012.
EndNote Horoub AMA, Nıcholson WK (January 1, 2021) $\mathcal{L}$-STABLE RINGS. International Electronic Journal of Algebra 29 29 63–94.
IEEE A. M. A. Horoub and W. K. Nıcholson, “$\mathcal{L}$-STABLE RINGS”, IEJA, vol. 29, no. 29, pp. 63–94, 2021, doi: 10.24330/ieja.852012.
ISNAD Horoub, Ayman M. A. - Nıcholson, W. K. “$\mathcal{L}$-STABLE RINGS”. International Electronic Journal of Algebra 29/29 (January 2021), 63-94. https://doi.org/10.24330/ieja.852012.
JAMA Horoub AMA, Nıcholson WK. $\mathcal{L}$-STABLE RINGS. IEJA. 2021;29:63–94.
MLA Horoub, Ayman M. A. and W. K. Nıcholson. “$\mathcal{L}$-STABLE RINGS”. International Electronic Journal of Algebra, vol. 29, no. 29, 2021, pp. 63-94, doi:10.24330/ieja.852012.
Vancouver Horoub AMA, Nıcholson WK. $\mathcal{L}$-STABLE RINGS. IEJA. 2021;29(29):63-94.