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Continuous Rings and Rings of Quotients

Published online by Cambridge University Press:  20 November 2018

S. S. Page
S. S. Page*
Affiliation:
Dept. of Mathematics, The University of British Columbia, Vancouver, B.C. V6T 1W5
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Throughout R will denote an associative ring with identity. Let Zℓ(R) be the left singular ideal of R. It is well known that Zℓ(R) = 0 if and only if the left maximal ring of quotients of R, Q(R), is Von Neumann regular. When Zℓ(R) = 0, q(R) is also a left self injective ring and is, in fact, the injective hull of R. A natural generalization of the notion of injective is the concept of left continuous as studied by Utumi [4]. One of the major obstacles to studying the relationships between Q(R) and R is a description of J(Q(R)), the Jacobson radical of Q(R). When a ring is left continuous, then its left singular ideal is its Jacobson radical. This facilitates the study of the cases when either Q(R) is continuous or R is continuous.

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 21 , Issue 3 , 01 September 1978 , pp. 319 - 324
Copyright
Copyright © Canadian Mathematical Society 1978

References

1. Gabriel, P. Des Catégories Abébinnes, Bull. Soc. Math. France, 90 (1962), 323-448.Google Scholar
2. Goldman, O. Rings and Modules of Quotients, J. Alg. 8 (1969), 10-47.Google Scholar
3. Page, S. S. Properties of Quotient Rings, Can. J. Math. 24 (1972) 1122-1128.Google Scholar
4. Utumi, Y. On Continuous Rings and Seĺf Infective Rings, Trans. Amer. Math. Soc. 118 (1965) 158-173.Google Scholar