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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

1. Cateforis, V. C. and Sandomierski, F. L., On modules of singular submodule zero, Can. J. Math. 28 (1971), 345–354.Google Scholar
2. Faith, C., Lectures on infective modules and quotient rings (Springer-Verlag, New York, 1967).Google Scholar
3. Johnson, R. E. and Wong, E. T., Self-injective rings, Can. Math. Bull. 2 (1959), 167–173.Google Scholar
4. Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Mass., 1966).Google Scholar
5. Ming, R. Yue Chi, A note on singular ideals, Töhoku Math. J. 21 (1969), 337–342.Google Scholar
6. Pierce, R. S., Modules over commutative regular rings, Amer. Math. Soc. Memoir No. 70 (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
7. Sandomierski, F. L., Nonsingular rings, Proc. Amer. Math. Soc. 19 (1968), 225–230.Google Scholar
8. Sandomierski, F. L., Semisimple maximal quotient rings, Trans. Amer. Math. Soc. 128 (1967), 112–120.Google Scholar