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Ideals in Topological Rings

Published online by Cambridge University Press:  20 November 2018

Bertram Yood
Bertram Yood*
Affiliation:
University of Oregon and Institute for Advanced Study
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We present here an investigation of the theory of one-sided ideals in a topological ring R. One of our aims is to discuss the question of "left" properties versus "right" properties. A problem of this sort is to decide if (a) all the modular maximal right ideals of R are closed if and only if all the modular maximal left ideals of R are closed. It is shown that this is the case if R is a quasi-Q-ring, that is, if R is bicontinuously isomorphic to a dense subring of a Q-ring (for the notion of a Q-ring see (6) or §2). All normed algebras are quasi-Q-rings. Also (a) holds if R is a semisimple ring with dense socle.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 28 - 45
Copyright
Copyright © Canadian Mathematical Society 1964

References

1. Arens, R., The space Loe and convex topological rings, Bull. Amer. Math. Soc, 52 (1946), 931–935.Google Scholar
2. Arnold, B. H., Rings of operators on vector spaces, Ann. of Math., 45 (1944), 24–49.Google Scholar
3. Bonsall, F. F. and Goldie, A. W., Annihilator algebras, Proc. London Math. Soc, 14 (1954), 154–167.Google Scholar
4. Jacobson, N., On the theory of primitive rings, Ann. of Math., 48 (1947), 8–21.Google Scholar
5. Jacobson, N., Structure of rings, Amer. Math. Soc. Coll. Publ., vol. 37 (Providence, 1956).Google Scholar
6. Kaplansky, I., Topological rings, Amer. J. Math., 69 (1947), 153–183.Google Scholar
7. Kaplansky, I., Locally compact rings, Amer. J. Math., 70 (1948), 447–459.Google Scholar
8. Kaplansky, I., Dual ringSf Ann. of Math., 49 (1948), 689–701.Google Scholar
9. Kaplansky, I., The structure of certain operator algebras, Trans. Amer. Math. Soc, 70 (1951), 219–255.Google Scholar
10. Kaplansky, I., Ring isomorphisms of Banach algebras, Can. J. Math., 6 (1954), 374–381.Google Scholar
11. Rickart, C. E., General theory of Banach algebras (New York, 1960).Google Scholar
12. Sawcrotnow, P. P., On a generalization of the notion of H*-algebra, Proc. Amer. Math. Soc, 8 (1957), 49–55.Google Scholar
13. Sawcrotnow, P. P. On the imbedding of a right complemented algebra into Ambrose's FL*-algebra, Proc Amer. Math. Soc, 8 (1957), 56–62.Google Scholar
14. E, I., Segal, The group algebra of a locally compact group, Trans. Amer. Math. Soc, 61 (1947), 69–105.Google Scholar
15. Smiley, M. F., Right annihilator algebras, Proc Amer. Math. Soc, 6 (1955), 698–701.Google Scholar
16. Wolfson, K. G., Annihilator rings, J. London Math. Soc, 81 (1956), 94–104.Google Scholar
17. Yood, B., Homomorphisms on normed algebras, Pacific J. Math., 8 (1958), 373–381.Google Scholar
18. Yood, B., Faithful *-representations of normed algebras, Pac J. Math., 10 (1960), 345–363.Google Scholar