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Injective Hulls of Torsion Free Modules

Published online by Cambridge University Press:  20 November 2018

J. Zelmanowitz
J. Zelmanowitz*
Affiliation:
Carnegie-Mellon University, Pittsburgh, Pennsylvania University of California, Santa Barbara, California
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In § 1, we begin with a basic theorem which describes a convenient embedding of a nonsingular left R-module into a complete direct product of copies of the left injective hull of R (Theorem 2). Several applications follow immediately. Notably, the injective hull of a finitely generated nonsingular left R-module is isomorphic to a direct sum of injective hulls of closed left ideals of R (Corollary 4). In particular, when R is left self-injective, every finitely generated nonsingular left R-module is isomorphic to a finite direct sum of injective left ideals (Corollary 6).

In § 2, where it is assumed for the first time that rings have identity elements, we investigate more generally the class of left R-modules which are embeddable in direct products of copies of the left injective hull Q of R. Such modules are called torsion free, and can also be characterized by the property that no nonzero element is annihilated by a dense left ideal of R (Proposition 12).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 6 , December 1971 , pp. 1094 - 1101
Copyright
Copyright © Canadian Mathematical Society 1971

References

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