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Bicommutators of Cofaithful, Fully Divisible Modules

Published online by Cambridge University Press:  20 November 2018

John A. Beachy
John A. Beachy*
Affiliation:
Northern Illinois University, DeKalb, Illinois
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We define below a notion for modules which is dual to that of faithful, and a notion of “fully divisible” which generalizes that of injectivity. We show that the bicommutator of a cofaithful, fully divisible left R-module is isomorphic to a subring of Qmax(R), the complete ring of left quotients of R.

In recent papers, Goldman [2] and Lambek [3] investigated rings of left quotients of a ring R constructed with respect to torsion radicals. It is known that every ring of left quotients of R is isomorphic to the bicommutator of an appropriate injective left R-module. We investigate below subrings of rings of quotients which are determined by radicals rather than torsion radicals, and show that any such ring can be constructed as the bicommutator of a fully divisible left R-module.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 2 , April 1971 , pp. 202 - 213
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Beachy, J. A., Generating and cogenerating structures, Trans. Amer. Math. Soc. (to appear).Google Scholar
2. Goldman, O., Rings and modules of quotients, J. Algebra 13 (1969), 10–47.Google Scholar
3. Lambek, J., Torsion theories, additive semantics, and rings of quotients, preprint (1970).Google Scholar
4. Lambek, J., Lectures on rings and modules (Blaisdell, Toronto, 1966).Google Scholar
5. Maranda, J.-M., Infective structures, Trans. Amer. Math. Soc. 110 (1964), 98–135.Google Scholar