Orthogonally complete alternative rings (Q1119730)

A subset X of a reduced (i.e. without non-zero nilpotent elements) ring A is called orthogonal if \(ab=0\) for all \(a,b\in X\) (a\(\neq b)\), and is orthogonally complete if every orthogonal subset of A has a supremum in A where the order is defined: \(a\leq b\) iff \(ab=a^ 2\). X is called boundable if \(ab(a-b)=0\) for every \(a,b\in X\), and A is complete if every boundable subset of A has in A a supremum. An alternative ring A is said to be von Neumann regular if for any element \(a\in A\) there exists an element \(b\in A\) that \(a=aba\). The main result is the following: A von Neumann regular reduced alternative ring is complete if and only if it is orthogonally complete.