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Projective prime ideals and localisation in pirings
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Chatters, A. W., Hajarnavis, C. R. and Lissaman, R. M.. (2001) Projective prime ideals and localisation in pirings. Journal of the London Mathematical Society, Vol.64 (No.1). pp. 112. ISSN 00246107

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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 PIring. Let P be a nonidempotent 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 PIring. Let M be a nonidempotent maximal ideal of R such that MR is projective. Then M has the left ARproperty 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:  00246107 
Date:  August 2001 
Volume:  Vol.64 
Number:  No.1 
Page Range:  pp. 112 
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 ARproperty in Noetherian PIrings ', Israel J. Math. 96 (1996) 115±140. 2. A. Braun and C. R. Hajarnavis, `A structure theorem for Noetherian PIrings 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, `Onesided 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 nonprime 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, `Noncommutative localized rings ', Séminaire d'Algèbre Paul Dubreil et MariePaule 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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