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Complete and Orthogonally Complete Rings

Published online by Cambridge University Press:  20 November 2018

W. D. Burgess  and
R. Raphael
W. D. Burgess
Affiliation:
University of Ottawa, Ottawa, Ontario
R. Raphael
Affiliation:
Concordia University, Montreal, Quebec
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This article continues the study of Abian's order on commutative semiprime rings (for such a ring R, the relation a ≦ b if and only if ab = a2” makes R into a partially ordered multiplicative semigroup). The aim, here, is to extend as far as possible the theorem of Brainerd and Lambek which says that the completion of a Boolean ring is its complete ring of quotients. Only certain subsets of a ring may have upper bounds (in any extension ring) and these are called boundable (the notion is due to Haines). A ring will be called complete if every boundable subset has a supremum. If R ⊂ S are (commutative semiprime) rings then S will be called a completion of R if S is complete and every element of S is the supremum of a subset of R.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 27 , Issue 4 , 01 August 1975 , pp. 884 - 892
Copyright
Copyright © Canadian Mathematical Society 1975

References

1. Abian, A., Direct product decomposition of commutative semisimple rings, Proc. Amer. Math. Soc. 24 (1970), 502–507.Google Scholar
2. Abian, A., Order relation in rings without nilpotent elements (typescript, 1973).Google Scholar
3. Banaschewski, B., Maximal rings of quotients of semi-simple commutative rings, Arch. Math. 16 (1965), 414–420.Google Scholar
4. Bergman, G. M., Hereditary commutative rings and centres of hereditary rings, Proc. London Math. Soc. 23 (1971), 214–236.Google Scholar
5. Bourbaki, N., Algèbre commutative, Chapitres 1 et 2, Fasc. 27 (Hermann, Paris, 1961).Google Scholar
6. Brainerd, B. and Lambek, J., On the ring of quotients of a Boolean ring, Can. Math. Bull. 2 (1959), 25–29.Google Scholar
7. Burgess, W. D. and Raphael, R., Abian s order relation and orthogonal completions for reduced rings, Pacific J. Math. 54 (1974), 55–64.Google Scholar
8. Chacron, M., Direct product of division rings and a paper of Abian, Proc. Amer. Math. Soc. 29 (1971), 259–262.Google Scholar
9. Clifford, A. A. and Preston, G. B., The algebraic theory of semigroups, Vol. II (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
10. Fine, N., Gillman, L., and Lambek, J., Rings of quotients of rings of functions (McGill University Press, Montreal, 1965).Google Scholar
11. Haines, D. C., Infective objects in the category of p-rings, Proc. Amer. Math. Soc. 42 (1974), 57–60.Google Scholar
12. Lambek, J., Lectures on rings and modules (Ginn, Waltham, Mass., 1966).Google Scholar
13. Raphael, R., Algebraic extensions of commutative regular rings, Can. J. Math. 22 (1970), 1133–1155.Google Scholar
14. Stenstrom, B., Rings and modules of quotients, Lecture Notes in Math. 237 (Springer, 1971).Google Scholar