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Localizations of essential extensions

Published online by Cambridge University Press:  20 January 2009

K. R. Goodearl  and
D. A. Jordan
K. R. Goodearl
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, Utah 84112, U.S.A.
D. A. Jordan
Affiliation:
Department of Pure Mathematics, University of Sheffield, Sheffield S3 7RH, England
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In an earlier paper [4] we considered the question of whether an injective module E over a noncommutative ring R remains injective after localization with respect to a denominator set X in R. A related question is whether, given an essential extension N of an R-module M, the localization N[X–1] must be an essential extension of M[X–1]. In [1] it is shown that if R is left noetherian and X is central in R, then localization at X preserves both injectivity and essential extensions of left R-modules and, hence, preserves injective hulls and minimal injective resolutions.

Information

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 31 , Issue 2 , June 1988 , pp. 243 - 247
Copyright
Copyright © Edinburgh Mathematical Society 1988

References

REFERENCES

1.Bass, H., Injective dimension in noetherian rings, Trans. Amer. Math. Soc. 102 (1962), 18–29.CrossRefGoogle Scholar
2.Chatters, A. W., The largest Ore set at a prime ideal: a special case, Bull. London Math. Soc. 18 (1986), 153–158.CrossRefGoogle Scholar
3.Cohn, P. M., Algebra, Vol. 2 (Wiley, London, 1977).Google Scholar
4.Goodearl, K. R. and Jordan, D. A., Localizations of injective modules, Proc. Edinburgh Math. Soc. 28 (1985), 289–299.CrossRefGoogle Scholar
5.Jategaonkar, A. V., Localization in Noetherian Rings (London Math. Soc. Lecture Note Series 98, Cambridge Univ. Press, Cambridge, 1986).CrossRefGoogle Scholar
6.Rinehart, G. S., Note on the global dimension of a certain ring, Proc. Amer. Math. Soc. 13 (1962), 341–346.CrossRefGoogle Scholar
7.Rotman, J. J., An Introduction to Homological Algebra (Academic Press, New York, 1979).Google Scholar
8.Sigurdsson, G., Links between prime ideals in differential operator rings, J. Algebra 102 (1986), 260–283.CrossRefGoogle Scholar
9.Stenström, B., Rings of Quotients (Springer-Verlag, Berlin, 1975).CrossRefGoogle Scholar