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

  • A. G. Heinicke (a1)

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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).

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References

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1. Goldman, O., Rings and modules of quotients, J. Algebra 13 (1969), 1047.
2. Lambek, J., Torsion theories, additive semantics, and rings of quotients, Lecture Notes in Mathematics, No. 177 (Springer-Verlag, New York, 1971).
3. Lambek, J. and Michler, G., The torsion theory at a prime ideal of a right noetherian ring (to appear).
4. Robson, J. C., Idealizers and hereditary noetherian prime rings (to appear).
5. Silver, L., Noncommutative localization and applications, J. Algebra 7 (1967), 4476.
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On the Ring of Quotients at a Prime Ideal of a Right Noetherian Ring

  • A. G. Heinicke (a1)

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