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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
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Abstract

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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