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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

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