On the Drazin index of regular elements

Pedro Patrício; António Costa

Open Mathematics (2009)

  • Volume: 7, Issue: 2, page 200-205
  • ISSN: 2391-5455

Abstract

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It is known that the existence of the group inverse a # of a ring element a is equivalent to the invertibility of a 2 a − + 1 − aa −, independently of the choice of the von Neumann inverse a − of a. In this paper, we relate the Drazin index of a to the Drazin index of a 2 a − + 1 − aa −. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.

How to cite

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Pedro Patrício, and António Costa. "On the Drazin index of regular elements." Open Mathematics 7.2 (2009): 200-205. <http://eudml.org/doc/269234>.

@article{PedroPatrício2009,
abstract = {It is known that the existence of the group inverse a # of a ring element a is equivalent to the invertibility of a 2 a − + 1 − aa −, independently of the choice of the von Neumann inverse a − of a. In this paper, we relate the Drazin index of a to the Drazin index of a 2 a − + 1 − aa −. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.},
author = {Pedro Patrício, António Costa},
journal = {Open Mathematics},
keywords = {Drazin inverse; Drazin index; Dedekind finite ring; Regular ring; regular ring},
language = {eng},
number = {2},
pages = {200-205},
title = {On the Drazin index of regular elements},
url = {http://eudml.org/doc/269234},
volume = {7},
year = {2009},
}

TY - JOUR
AU - Pedro Patrício
AU - António Costa
TI - On the Drazin index of regular elements
JO - Open Mathematics
PY - 2009
VL - 7
IS - 2
SP - 200
EP - 205
AB - It is known that the existence of the group inverse a # of a ring element a is equivalent to the invertibility of a 2 a − + 1 − aa −, independently of the choice of the von Neumann inverse a − of a. In this paper, we relate the Drazin index of a to the Drazin index of a 2 a − + 1 − aa −. We give an alternative characterization when considering matrices over an algebraically closed field. We close with some questions and remarks.
LA - eng
KW - Drazin inverse; Drazin index; Dedekind finite ring; Regular ring; regular ring
UR - http://eudml.org/doc/269234
ER -

References

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  1. [1] Castro González N., Additive perturbation results for the Drazin inverse, Linear Algebra Appl., 2005, 397, 279–297 http://dx.doi.org/10.1016/j.laa.2004.11.001[Crossref][WoS] Zbl1071.15003
  2. [2] Cline R.E., An application of representation of a matrix, MRC Technical Report, 592, 1965 
  3. [3] Drazin M.P., Pseudo inverses in associative rings and semigroups, Amer. Math. Monthly, 1958, 65, 506–514 http://dx.doi.org/10.2307/2308576[Crossref] Zbl0083.02901
  4. [4] Hartwig R.E., Luh J., On finite regular rings, Pacific J. Math., 1977, 69, 73–95 Zbl0326.16012
  5. [5] Hartwig R.E., Patricio P., A note on power bounded matrices, preprint 
  6. [6] Hartwig R.E., Shoaf J., Group inverses and Drazin inverse of bidiagonal and triangular Toeplitz matrices, J. Austral. Math. Soc. Ser. A, 1977, 24, 10–34 http://dx.doi.org/10.1017/S1446788700020036[Crossref] Zbl0372.15003
  7. [7] Hartwig R.E., Wang G., Wei Y., Some additive results on Drazin inverses, Linear Algebra Appl., 2001, 322, 207–217 http://dx.doi.org/10.1016/S0024-3795(00)00257-3[Crossref] Zbl0967.15003
  8. [8] Patricio P., Puystjens R., Generalized invertibility in two semigroups of a ring, Linear Algebra Appl., 2004, 377, 125–139 http://dx.doi.org/10.1016/j.laa.2003.08.004[Crossref] 
  9. [9] Puystjens R., Hartwig R.E., The group inverse of a companion matrix, Linear and Multilinear Algebra, 1997, 43, 137–150 http://dx.doi.org/10.1080/03081089708818521[Crossref] Zbl0890.15003
  10. [10] Roman S., Advanced linear algebra, Graduate Texts in Mathematics 135, Springer, New York, 2005 Zbl1085.15001
  11. [11] Schmoeger C., On Fredholm properties of operator products, Math. Proc. R. Ir. Acad. 103A, 2003, 2, 203–208 http://dx.doi.org/10.3318/PRIA.2003.103.2.203[Crossref] Zbl1084.47009

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