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 a≠b 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 orthogonalcompletion. 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.
