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 of R 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 QF(R) of R with
respect to an idempotent filter F of dense right ideals of R. Furthermore, it is shown
in Proposition 5 that QF(R) is an orthogonal extension of R if and only if for
every q ∈ QF(R), there exists a maximal orthogonal subset {ei: i ∈ I} of
idempotents of Q(R) such that q maps (by left multiplication) the right
R-submodule of Q(R) generated by q−1R ∪{ei: i ∈ I} into R. Also an orthogonal
extension QF(R) is an orthogonal completion of R if and only if for every
R-submodule MR of Q(R)R generated by a maximal orthogonal subset of
idempotents of Q(R) and for every f ∈HomR(M,R) there exists a q ∈ QF(R)
such that f(m) = qm for every m ∈ M (Proposition 6). Thus we obtain a
necessary and sufficient condition for a reduced ring to have an orthogonal
completion without any reference to its idempotent which improves earlier
known results derived by Burgess and Raphael. By examples we show that
reduced rings without proper idempotents may also have an orthogonal
completion.
