Vol. 108, No. 2, 1983

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On orthogonal completion of reduced rings

R. K. Rai

Vol. 108 (1983), No. 2, 385–396

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 q1R ∪{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.

Mathematical Subject Classification 2000
Primary: 16A86, 16A86
Secondary: 06F25
Received: 8 May 1980
Revised: 20 May 1982
Published: 1 October 1983
R. K. Rai