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Localizations of injective modules

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, USA
D. A. Jordan
Affiliation:
Department of Pure Mathematics, University OF Sheffield, Sheffield S3 7RH, England
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The question of whether an injective module E over a noncommutative noetherian ring R remains injective after localization with respect to a denominator set X⊆R is addressed. (For a commutative noetherian ring, the answer is well-known to be positive.) Injectivity of the localization E[X-1] is obtained provided either R is fully bounded (a result of K. A. Brown) or X consists of regular normalizing elements. In general, E [X-1] need not be injective, and examples are constructed. For each positive integer n, there exists a simple noetherian domain R with Krull and global dimension n+1, a left and right denominator set X in R, and an injective right R-module E such that E[X-1 has injective dimension n; moreover, E is the injective hull of a simple module.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 28 , Issue 3 , October 1985 , pp. 289 - 299
Copyright
Copyright © Edinburgh Mathematical Society 1985

References

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