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Projective prime ideals and localisation in pi-rings

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Chatters, A. W., Hajarnavis, C. R. and Lissaman, R. M.. (2001) Projective prime ideals and localisation in pi-rings. Journal of the London Mathematical Society, Vol.64 (No.1). pp. 1-12. ISSN 0024-6107

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Official URL: http://dx.doi.org/10.1017/S0024610701002125

Abstract

The results here generalise [2, Proposition 4.3] and [9, Theorem 5.11]. We shall prove the following.

THEOREM A. Let R be a Noetherian PI-ring. Let P be a non-idempotent prime ideal of R such that PR is projective. Then P is left localisable and RP is a prime principal left and right ideal ring.

We also have the following theorem.

THEOREM B. Let R be a Noetherian PI-ring. Let M be a non-idempotent maximal ideal of R such that MR is projective. Then M has the left AR-property and M contains a right regular element of R.

Item Type:

Journal Article

Subjects:

Q Science > QA Mathematics

Divisions:

Faculty of Science > Mathematics

Library of Congress Subject Headings (LCSH):

Algebra, Homological, Associative rings, Associative algebras

Journal or Publication Title:

Journal of the London Mathematical Society

Publisher:

Cambridge University Press

ISSN:

0024-6107

Official Date:

August 2001

Dates:

Date	Event
August 2001	Published

Volume:

Vol.64

Number:

No.1

Page Range:

pp. 1-12

Identification Number:

10.1017/S0024610701002125

Status:

Peer Reviewed

Access rights to Published version:

Open Access

Funder:

London Mathematical Society (LMS)

References:

1. A. Braun, `Localization, completion and the AR-property in Noetherian PI-rings ', Israel J. Math. 96 (1996) 115±140.
2. A. Braun and C. R. Hajarnavis, `A structure theorem for Noetherian PI-rings with global dimension 2', J. Algebra 215 (1999) 248±289.
3. A. Braun and R. B. Warfield Jr, `Symmetry and localization in Noetherian prime PI rings ', J. Algebra 118 (1988) 322±335.
4. A. W. Chatters and C. R. Hajarnavis, Rings with chain conditions, Research Notes in Mathematics 44 (Pitman, London, 1980).
5. K. R. Goodearl and R. B. Warfield, An introduction to noncommutative Noetherian rings, London Mathematical Society Student Texts 16 (Cambridge University Press, Cambridge, 1989).
6. C. R. Hajarnavis, `One-sided invertibility and localisation II ', Glasgow J. Math. 37 (1995) 15±19.
7. C. R. Hajarnavis and T. H. Lenagan, `Localization in Asano orders', J. Algebra 21 (1972) 441±449.
8. T. H. Lenagan, `Artinian ideals in Noetherian rings', Proc. Amer. Math. Soc. 51 (1975) 499±500.
9. R. M. Lissaman, `Internal characterisations of non-prime Dedekind orders', Comm. Algebra 27 (1999) 4569±4585.
10. J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Pure and Applied Mathematics (John Wiley, New York, 1987).
11. R. B. Warfield, `Non-commutative localized rings ', Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin : Proceedings, Paris, 1985 (37ème Année), Lecture Notes in Mathematics 1220 (Springer, 1986) 178±200.

URI:

http://wrap.warwick.ac.uk/id/eprint/805

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