Right simple injective FGF-ring

A ring R is called right FGF-ring if every finitely generated right R-module embeds in a free (projective) . A ring is called right simple-injective if RR is simple R-injective, that is, if I is a right ideal of R and : I ! R is an R-morphismwith simple image, then (x) = c:x, is leftmultiplication by an element c 2 R. There is a conjecture due to Carl Faith which asserts that every right FGF-ring is a Quasi-Frobenius ring (QF). In this paper we establish the conjecture in case that the ring is a simple injective ring by showing that the right simple-injective FGF ring is a right self- injective.