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On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

Published online by Cambridge University Press:  20 November 2018

A. G. Heinicke
A. G. Heinicke*
Affiliation:
University of Western Ontario, London, Ontario
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J. Lambek and G. Michler [3] have initiated the study of a ring of quotients RP associated with a two-sided prime ideal P in a right noetherian ring R. The ring RP is the quotient ring (in the sense of [1]) associated with the hereditary torsion class τ consisting of all right R-modules M for which HomR(M, ER(R/P)) = 0, where ER(X) is the injective hull of the R-module X.

In the present paper, we shall study further the properties of the ring RP. The main results are Theorems 4.3 and 4.6. Theorem 4.3 gives necessary and sufficient conditions for the torsion class associated with P to have property (T), as well as some properties of RP when these conditions are indeed satisfied, while Theorem 4.6 gives necessary and sufficient conditions for R to satisfy the right Ore condition with respect to (P).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 4 , August 1972 , pp. 703 - 712
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Goldman, O., Rings and modules of quotients, J. Algebra 13 (1969), 10–47.Google Scholar
2. Lambek, J., Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, No. 177 (Springer-Verlag, New York, 1971).Google Scholar
3. Lambek, J. and Michler, G., The torsion theory at a prime ideal of a right noetherian ring (to appear).Google Scholar
4. Robson, J. C., Idealizers and hereditary noetherian prime rings (to appear).Google Scholar
5. Silver, L., Noncommutative localization and applications, J. Algebra 7 (1967), 44–76.Google Scholar