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On the Complete Ring of Quotients

Published online by Cambridge University Press:  20 November 2018

Kwangil Koh
Kwangil Koh*
Affiliation:
North Carolina State University, Raleigh, North Carolina27607
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In [2: p. 415], P. Gabriel proves that if R is a ring with 1 and S is a non-empty multiplicative set such that 0∉S, then S-1R exists if and only if for every pair (a, s)∈R×S, there is a pair (b, t)∈R×S such that at=sb and if s1a=0 for some s1 in S then as2=0 for some s2 in S. The purpose of this note is to give a self contained elementary proof of Gabriel’s result.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 17 , Issue 2 , 01 June 1974 , pp. 285 - 288
Copyright
Copyright © Canadian Mathematical Society 1974

References

1. Findlay, G. D. and Lamhek, J., A generalized ring of quotients I, Canadian Math. Bull., Vol. l, no. 2, May 1958.Google Scholar
2. Gabriel, P., Des categories abeliennes, Bull. Soc. Math., France 90 (1962), 325-448.Google Scholar
3. Johnson, R. E., The extended centralizer of a ring over a module, Proc. Amer. Math. Soc. 2 (1951), 891-895.Google Scholar