Vol. 54, No. 1, 1974

Download this article
Download this article. For screen
For printing
Recent Issues
Vol. 307: 1  2
Vol. 306: 1  2
Vol. 305: 1  2
Vol. 304: 1  2
Vol. 303: 1  2
Vol. 302: 1  2
Vol. 301: 1  2
Vol. 300: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
Abian’s order relation and orthogonal completions for reduced rings

Walter D. Burgess and Robert Raphael

Vol. 54 (1974), No. 1, 55–64

Chacron has shown that, in a ring R, the relation a b iff ab = a2”, first studied by Abian, is an order relation iff R is reduced (has no nilpotent elements). Let R be a reduced ring with 1, a set X in R is orthogonal if ab = 0 for all ab in X and R is orthogonally complete if every orthogonal set in R has a supremum with respect to ”. A strongly regular ring is shown to be right (and left) self-injective iff it is orthogonally complete. If R S are reduced rings, S is an orthogonal extension of R if every element of S is the supremum of an orthogonal set in R; an orthogonal extension which is complete is an orthogonal completion. Completions are unique if they exist. An example shows that not all reduced rings have completions but if R is strongly regular, its complete ring of quotients, Q(R), is its completion. Further, if R is reduced, Baer and such that Q(R) is strongly regular then R has a completion which is a partial ring of quotients.

Mathematical Subject Classification
Primary: 16A56
Secondary: 06A70
Received: 5 March 1973
Published: 1 September 1974
Walter D. Burgess
Robert Raphael