Hostname: page-component-7479d7b7d-fwgfc Total loading time: 0 Render date: 2024-07-13T11:31:17.466Z Has data issue: false hasContentIssue false
  • English
  • Français

Semicritical Rings and the Quotient Problem

Published online by Cambridge University Press:  20 November 2018

Karl A. Kosler
Karl A. Kosler*
Affiliation:
St. Norbert CollegeDe Pere, Wisconsin54115
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.

The purpose of this paper is to examine the relationship between the quotient problem for right noetherian nonsingular rings and the quotient problem for semicritical rings. It is shown that a right noetherian nonsingular ring R has an artinian classical quotient ring iff certain semicritical factor rings R/Ki, i = 1,…,n, possess artinian classical quotient rings and regular elements in R/Ki lift to regular elements of R for all i. If R is a two sided noetherian nonsingular ring, then the existence of an artinian classical quotient ring is equivalent to each R/Ki possessing an artinian classical quotient ring and the right Krull primes of R consisting of minimal prime ideals. If R is also weakly right ideal invariant, then the former condition is redundant. Necessary and sufficient conditions are found for a nonsingular semicritical ring to have an artinian classical quotient ring.

Keywords

16A4616A6616A55
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 27 , Issue 2 , 01 June 1984 , pp. 160 - 170
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Boyle, A. K. and Feller, E. H., Semicritical Modules and k-Primitive Rings, in Module Theory, Springer-Verlag Lecture Notes No. 700, 1979, 57–74.Google Scholar
2. Boyle, A. K. and Feller, E. H., α-Coprimitive Ideals and α-Indecomposable Infectives, Comm. in Alg., Vol. 8, 1980, 1151–1167.Google Scholar
3. Boyle, A. K. and Feller, E. H., α-Injectives and the Semicritical Socle Series, Preprint.Google Scholar
4. Boyle, A. K.; The Large Condition for Rings with Krull Dimension, Proc. A.M.S., Vol. 27, 1978, 27–32.Google Scholar
5. Feller, E. H., Rings where the Annihilator of α-Critical Modules are Prime Ideals, Pac. J. Math., Vol. 93, 1981, 299–306.Google Scholar
6. Goldie, A. W. and Krause, G.; Artinian Quotient Rings of Ideal Invariant Noetherian Rings, J. Alg., Vol. 63, No. 2, 1980, 374–388.Google Scholar
7. Goldie, A. W.; The Structure of Noetherian Rings, in Lectures on Rings and Modules, Springer-Verlag Lecture Notes No. 246, 1972, 213–321.Google Scholar
8. Goodearl, K. R.; Ring Theory, Marcel-Dekker, 1976.Google Scholar
9. Gordon, R. and Robson, J. C.; Krull Dimension, Memoirs A.M.S., No. 133, 1973.Google Scholar
10. Muller, B. J.; The Quotient Problem for Noetherian Rings, Can. J. Math., Vol. 33, No. 3, 1981, 734–748.Google Scholar
11. Small, L. W.; Orders in Artinian Rings, J. Alg., Vol. 4, 1966, 13–41.Google Scholar