Hostname: page-component-586b7cd67f-2plfb Total loading time: 0 Render date: 2024-11-22T05:02:50.909Z Has data issue: false hasContentIssue false
  • English
  • Français

THE LINK BETWEEN REGULARITY AND STRONG-PI-REGULARITY

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 August 2010

PEDRO PATRÍCIO  and
R. E. HARTWIG
PEDRO PATRÍCIO*
Affiliation:
Departamento de Matemática e Aplicações, Universidade do Minho, 4710-057 Braga, Portugal (email: [email protected])
R. E. HARTWIG
Affiliation:
Department of Mathematics, N.C.S.U., Raleigh, NC 27695-8205, USA (email: [email protected])
*
✉For correspondence; e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is shown that if all powers of a ring element a are regular, then a is strongly pi-regular exactly when a suitable word in the powers of a and their inner inverses is a unit.

Keywords

Drazin inversestrongly pi-regularDrazin index

MSC classification

Secondary: 15A09: Matrix inversion, generalized inverses
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 89 , Issue 1 , August 2010 , pp. 17 - 22
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2010

Footnotes

This research received financial support from the Research Centre of Mathematics of the University of Minho (CMAT) through the FCT Pluriannual Funding Program.

References

[1]Castro-González, N., Mendes Araújo, C. and Patrício, P., ‘A note on generalized inverses of a sum in rings’, Bull. Aust. Math. Soc. 82 (2010), 156–164.CrossRefGoogle Scholar
[2]Drazin, M. P., ‘Pseudo-inverses in associative rings and semigroups’, Amer. Math. Monthly 65 (1958), 506–514.CrossRefGoogle Scholar
[3]Hall, F. J. and Hartwig, R. E., ‘Algebraic properties of governing matrices used in Cesàro–Neumann iterations’, Rev. Roumaine Math. Pures Appl. 26(7) (1981), 959–978.Google Scholar
[4]Hartwig, R. E., ‘More on the Souriau–Frame algorithm and the Drazin inverse’, SIAM J. Appl. Math. 31(1) (1976), 42–46.CrossRefGoogle Scholar
[5]Patrício, P. and Veloso da Costa, A., ‘On the Drazin index of regular elements’, Cent. Eur. J. Math. 7(2) (2009), 200–205.Google Scholar
[6]Puystjens, R. and Gouveia, M. C., ‘Drazin invertibility for matrices over an arbitrary ring’, Linear Algebra Appl. 385 (2004), 105–116.CrossRefGoogle Scholar
[7]Puystjens, R. and Hartwig, R. E., ‘The group inverse of a companion matrix’, Linear Multilinear Algebra 43(1–3) (1997), 137–150.CrossRefGoogle Scholar