Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-27T19:37:26.892Z Has data issue: false hasContentIssue false
  • English
  • Français

Left orders in regular rings with minimum condition for principal one-sided ideals

Published online by Cambridge University Press:  24 October 2008

Pham Ngoc Ánh  and
László Márki
Pham Ngoc Ánh
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary
László Márki
Affiliation:
Mathematical Institute, Hungarian Academy of Sciences, H-1364 Budapest, Pf. 127, Hungary
Get access Rights & Permissions [Opens in a new window]

Extract

Based on ideas from semigroup theory, Fountain and Gould [2, 3, 4] introduced a notion of order in a ring which need not have an identity. In some important cases of rings with identity, e.g. if the larger ring is a semisimple artinian ring, this notion coincides with the classical one. The most important result of Fountain and Gould (see [4]) is a Goldie-like characterization of two-sided orders in a regular ring with minimum condition on principal one-sided ideals. In addition, for the same class of rings, a generalization of the Faith–Utumi theorem has been proved by Gould and Petrich[7]. The methods of these papers seem not to work for one-sided orders.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 109 , Issue 2 , March 1991 , pp. 323 - 333
Copyright
Copyright © Cambridge Philosophical Society 1991

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Ánh, P. N. and Márki, L.. Rees matrix rings. J. Algebra 81 (1983), 340–369.CrossRefGoogle Scholar
[2]Fountain, J. and Gould, V.. Orders in rings without identity. Comm. Algebra. (To appear.)Google Scholar
[3]Fountain, J. and Gould, V.. Straight left orders in rings. (Preprint, 1988.)Google Scholar
[4]Fountain, J. and Gould, V.. Orders in regular rings with minimum condition for principal right ideals. (Preprint, 1989.)Google Scholar
[5]Goldie, A. W.. Semi-prime rings with maximum condition. Proc. London Math. Soc. (3), 10 (1960), 201–220.CrossRefGoogle Scholar
[6]Goldie, A. W.. The structure of Noetherian rings. In Lectures on Rings and Modules, Lecture Notes in Math. vol. 246 (Springer-Verlag, 1972), pp. 213–321.CrossRefGoogle Scholar
[7]Gould, V. and Petrich, M.. A new approach to orders in simple rings with minimal one-sided ideals. Semigroup Forum 41 (1990), 267–290.CrossRefGoogle Scholar
[8]Jacobson, X.. Structure of Rings, revised edition. Amer. Math. Soc. Colloquium Publications, no. 37 (American Mathematical Society, 1964).Google Scholar
[9]Loi, N. V.. Semiprime rings with d.c.c. on principal bi-ideals. Period. Math. Hungar. 17 (1986), 71–75.CrossRefGoogle Scholar
[10]Stenström, B.. Rings of Quotients. Grundlehren der Math. Wiss. no. 217 (Springer-Verlag, 1975).CrossRefGoogle Scholar