On a theorem of Fuller Original Research Article

Let e and f be primitive idempotents in a semiperfect ring R with radical J. Suppose that (i) fR/fJ congruent with Soc(eRR), which is essential in eRR; and (ii) Re/Je congruent with Soc(RRf), which is essential in RRf. We prove that the following are equivalent: (1) both eReeR and RffRf are linearly compact; (2) both eRR and RRf are injective; (3) eRR is injective and RffRf is linearly compact; and (4) eReeR is linearly compact and RRf is injective. This generalizes an old theorem of Fuller and a recent theorem of Baba and Oshiro.