On orthogonal completion of reduced rings

Abstract

It was proved by the author earlier that every orthogonal extension of a reduced ring R is a subring of Q(R), the maximal two sided ring of quotients oiR and the orthogonal completion of R, if it exists, is unique upto an isomorphism. Here, in Theorem 2, we prove that the orthogonal completion of R, if it exists, is a ring of right quotients Q F (R) of R with respect to an idempotent filter F of dense right ideals of R. Introduction. Abian [2] showed that the canonical order relation ' < ' of Boolean rings can be defined for reduced rings R (a ring with no nonzero nilpotent element) by writing a < b if ab -a 2 and this order relation makes R into a partially ordered multiplicative semigroup. Reduced rings under this relation ' < ' were studied by Abian [1] and Chacron [5] to characterise the direct produce of integral domains, division rings and fields. Their studies involved the concepts of orthogonal completeness and orthogonal completion of reduced rings. These two concepts, on their own merit, were studied by Burgess, Raphael and Stephenson [3], [4], [11]. They proved that reduced rings which have enough idempotents (/-dense) or satisfy certain chain conditions have an orthogonal completion. In this paper we shall provide a necessary and sufficient condition for a reduced ring to have an orthogonal completion.

RATIONALLY ℵα-COMPLETE COMMUTATIVE RINGS OF QUOTIENTS

“…Rings of quotients have been regarded by several authors as order completions (e.g., [2,3,13]). In [2], it is shown that the usual order relation on any Boolean ring R is complete if and only if R = Q(R).…”