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On I-finite left quasi-duo rings

Year 2022, Volume: 31 Issue: 31, 161 - 202, 17.01.2022
https://doi.org/10.24330/ieja.1058427

Abstract

A ring is called left quasi-duo (left QD) if every maximal left ideal is a
right ideal, and it is called I-finite if it contains no infinite orthogonal
set of idempotents. It is shown that a ring is I-finite and left QD if and
only if it is a generalized upper-triangular matrix ring with all diagonal
rings being division rings except the lower one, which is either a division
ring or it is I-finite, left QD and left `soclin' (left QDS). Here a ring is
called left soclin if each simple left ideal is nilpotent. The left QDS
rings are shown to be finite direct products of indecomposable left QDS
rings, in each of which 1=f1++fm1=f1+⋯+fm where the fifi are
orthogonal primitive idempotents, with fkflfk≈fl for all k,l,k,l,
and is the block equivalence on {f1,,fm}.{f1,…,fm}.

A ring is shown to be left soclin if and only if every maximal left ideal is
left essential, if and only if the left socle is contained in the left
singular ideal. These left soclin rings are proved to be a Morita invariant
class; and if a ring is semilocal and non-semisimple, then it is left soclin
if and only if the Jacobson radical is essential as a left ideal..

Left quasi-duo elements are defined for any ring and shown to constitute a
subring containing the centre and the Jacobson radical of the ring. The
`width' of any left QD ring is defined and applied to characterize the
semilocal left QD rings, and to clarify the semiperfect case..

References

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  • R. Brauer, Harvard Lectures on the Wedderburn Theorems, 1950.
  • W. Burgess and W. Stephenson, Rings all of whose Pierce stalks are local, Canad. Math. Bull., 22(2) (1979), 159-164.
  • V. P. Camillo and H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra, 22(12) (1994), 4737-4749.
  • P. Crawley and B. Jonsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math., 14 (1964), 797-855.
  • M. P. Drazin, Rings with central idempotent or nilpotent elements, Proc. Edinburgh Math. Soc., 9 (1958), 157-165.
  • E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc., 89 (1958), 79-91.
  • K. R. Goodearl, Ring Theory, Nonsingular Rings and Modules, Marcel Dekker, Inc., New York-Basel, 1976.
  • C. Huh, S. H. Jang, C. O. Kim and Y. Lee, Rings whose maximal one-sided ideals are two-sided, Bull. Korean Math. Soc., 39(3) (2002), 411-422.
  • N. Jacobson, Some remarks on one-sided inverses, Proc. Amer. Math. Soc., 1 (1950), 352-355.
  • C. O. Kim, H. K. Kim and S. H. Jang, A study on quasi-duo rings, Bull. Korean Math. Soc., 36 (1999), 579-588.
  • N. K. Kim and Y. Lee, On right quasi-duo rings which are $\pi$-regular, Bull. Korean Math. Soc., 37 (2000), 217-227.
  • T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.
  • T. Y. Lam and A. S. Dugas, Quasi-duo rings and stable range descent, J. Pure Appl. Algebra, 195 (2005), 243-259.
  • J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, Ltd., Chichester, 1987.
  • W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
  • W. K. Nicholson, Very semisimple modules, Amer. Math. Monthly, 104(2) (1997), 159-162.
  • W. K. Nicholson and E. S´anchez Campos, Rings with the dual of the isomorphism theorem, J. Algebra, 271 (2004), 391-406.
  • W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics, 158, Cambridge University Press, Cambridge, 2003.
  • L. Rowen, Ring Theory, Volume I, Academic Press, Inc., Boston, MA, 1988.
  • J. C. Shepherdson, Inverses and zero-divisors in matrix rings, Proc. London Math. Soc., (3)1 (1951), 71-85.
  • R. Warfield, Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31-36.
  • H.-P. Yu, On quasi-duo rings, Glasgow Math. J., 37 (1995), 21-31.
Year 2022, Volume: 31 Issue: 31, 161 - 202, 17.01.2022
https://doi.org/10.24330/ieja.1058427

Abstract

References

  • F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Second Edition, Springer-Verlag, New York, 1992.
  • R. Brauer, Harvard Lectures on the Wedderburn Theorems, 1950.
  • W. Burgess and W. Stephenson, Rings all of whose Pierce stalks are local, Canad. Math. Bull., 22(2) (1979), 159-164.
  • V. P. Camillo and H.-P. Yu, Exchange rings, units and idempotents, Comm. Algebra, 22(12) (1994), 4737-4749.
  • P. Crawley and B. Jonsson, Refinements for infinite direct decompositions of algebraic systems, Pacific J. Math., 14 (1964), 797-855.
  • M. P. Drazin, Rings with central idempotent or nilpotent elements, Proc. Edinburgh Math. Soc., 9 (1958), 157-165.
  • E. H. Feller, Properties of primary noncommutative rings, Trans. Amer. Math. Soc., 89 (1958), 79-91.
  • K. R. Goodearl, Ring Theory, Nonsingular Rings and Modules, Marcel Dekker, Inc., New York-Basel, 1976.
  • C. Huh, S. H. Jang, C. O. Kim and Y. Lee, Rings whose maximal one-sided ideals are two-sided, Bull. Korean Math. Soc., 39(3) (2002), 411-422.
  • N. Jacobson, Some remarks on one-sided inverses, Proc. Amer. Math. Soc., 1 (1950), 352-355.
  • C. O. Kim, H. K. Kim and S. H. Jang, A study on quasi-duo rings, Bull. Korean Math. Soc., 36 (1999), 579-588.
  • N. K. Kim and Y. Lee, On right quasi-duo rings which are $\pi$-regular, Bull. Korean Math. Soc., 37 (2000), 217-227.
  • T. Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.
  • T. Y. Lam and A. S. Dugas, Quasi-duo rings and stable range descent, J. Pure Appl. Algebra, 195 (2005), 243-259.
  • J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, John Wiley & Sons, Ltd., Chichester, 1987.
  • W. K. Nicholson, Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278.
  • W. K. Nicholson, Very semisimple modules, Amer. Math. Monthly, 104(2) (1997), 159-162.
  • W. K. Nicholson and E. S´anchez Campos, Rings with the dual of the isomorphism theorem, J. Algebra, 271 (2004), 391-406.
  • W. K. Nicholson and M. F. Yousif, Quasi-Frobenius Rings, Cambridge Tracts in Mathematics, 158, Cambridge University Press, Cambridge, 2003.
  • L. Rowen, Ring Theory, Volume I, Academic Press, Inc., Boston, MA, 1988.
  • J. C. Shepherdson, Inverses and zero-divisors in matrix rings, Proc. London Math. Soc., (3)1 (1951), 71-85.
  • R. Warfield, Exchange rings and decompositions of modules, Math. Ann., 199 (1972), 31-36.
  • H.-P. Yu, On quasi-duo rings, Glasgow Math. J., 37 (1995), 21-31.
There are 23 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 17, 2022
Published in Issue Year 2022 Volume: 31 Issue: 31

Cite

APA Horoub, A. M. A., & Nıcholson, W. K. (2022). On I-finite left quasi-duo rings. International Electronic Journal of Algebra, 31(31), 161-202. https://doi.org/10.24330/ieja.1058427
AMA Horoub AMA, Nıcholson WK. On I-finite left quasi-duo rings. IEJA. January 2022;31(31):161-202. doi:10.24330/ieja.1058427
Chicago Horoub, Ayman M. A., and W. K. Nıcholson. “On I-Finite Left Quasi-Duo Rings”. International Electronic Journal of Algebra 31, no. 31 (January 2022): 161-202. https://doi.org/10.24330/ieja.1058427.
EndNote Horoub AMA, Nıcholson WK (January 1, 2022) On I-finite left quasi-duo rings. International Electronic Journal of Algebra 31 31 161–202.
IEEE A. M. A. Horoub and W. K. Nıcholson, “On I-finite left quasi-duo rings”, IEJA, vol. 31, no. 31, pp. 161–202, 2022, doi: 10.24330/ieja.1058427.
ISNAD Horoub, Ayman M. A. - Nıcholson, W. K. “On I-Finite Left Quasi-Duo Rings”. International Electronic Journal of Algebra 31/31 (January 2022), 161-202. https://doi.org/10.24330/ieja.1058427.
JAMA Horoub AMA, Nıcholson WK. On I-finite left quasi-duo rings. IEJA. 2022;31:161–202.
MLA Horoub, Ayman M. A. and W. K. Nıcholson. “On I-Finite Left Quasi-Duo Rings”. International Electronic Journal of Algebra, vol. 31, no. 31, 2022, pp. 161-02, doi:10.24330/ieja.1058427.
Vancouver Horoub AMA, Nıcholson WK. On I-finite left quasi-duo rings. IEJA. 2022;31(31):161-202.