Orthogonally complete rings
Canadian mathematical bulletin
Introduction. In this note we continue the study of Abian's order for reduced rings initiated in papers such as ,  , , . A simple proof is given of Abian's result that taking suprema commutes with ring multiplication. The properties of orthogonally complete rings and of rings satisfying chain conditions with respect to Abian's order are investigated. Finally those rings JR for which R[x] and JR [[x]] are orthogonally complete are characterized. These results provide interesting
... de interesting examples and counterexamples in the study of Abian's order relation. Terminology. A ring R is said to be reduced if it contains no non-zero nilpotent elements. Throughout this note we consider only reduced rings with a 1, unless otherwise stated. We recall some elementary facts about such rings. If a,beR and ab = 0, then ba = 0. Hence the left and right annihilators of any subset S ç R form a two-sided ideal which we denote by Ann(S). In particular for any aeR, Ann(a)fl JRa = 0. For other standard terminology the reader is referred to  . §1. Any reduced ring can be written as a subdirect product of integral domains (not necessarily commutative). Since any integral domain can be ordered by decreeing that c ^ 0 for any c, this induces an ordering on JR given by a ^ b if a 2 = ab (see  and  for details). With respect to this ordering we can discuss upper and lower bounds, suprema and infima of subsets of JR in the usual manner.