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A Generalized Ring of Quotients I

Published online by Cambridge University Press:  20 November 2018

G.D. Findlay  and
J. Lambek
G.D. Findlay
Affiliation:
McGill University
J. Lambek
Affiliation:
McGill University
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According to a well-known theorem of algebra [3, p.47], an integral domain can be embedded in a field, called its field of quotients. Every freshman is familiar with the simplest form of this theorem concerning the integers and the rational numbers. Many generalisations have been given in which a "ring of quotients" is constructed for a given ring [e.g. 6, 7, 11, 12, 13]. Of course, we cannot expect a ring of quotients to be a field, or even a skew field. As Malcev [9] has shown, there even exist rings without proper divisors of zero which cannot be embedded isomorphically in a skew field. The most recent construction, by Utumi [12], gives a ring of quotients for any ring with zero left annihilator. We show in this paper that this construction can be extended to arbitrary rings, in fact, to arbitrary modules. The method used is more abstract: a fundamental relation between rings, defined by Utumi in terms of their elements, is here replaced by a corresponding relation between modules, defined by means of homomorphisms.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 1 , Issue 2 , May 1958 , pp. 77 - 85
Copyright
Copyright © Canadian Mathematical Society 1958

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