Vol. 54, No. 1, 1974

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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