ALMOST QF RINGS WITH J(3)=0

In this paper we always assume that R is a two-sided artinian ring with identity. In [3] we have defined right almost QF rings and showed that those rings coincided with rings satisfying (*)* in [2], which K. Oshiro [5] called co-H rings. We shall show in Section 2 that right almost QF rings are nothing but direct sums of serial rings and QF rings, provided J3 = 0. Further in Section 5 we show that if R is a two-sided almost QF ring and 1 = e1 + e2 + e3, then R has the above structure, provided J4 = 0, where {e(t)} is a complete set of mutually orthogonal primitive idempotents. Moreover if 1 = e1 + e2 + e3 + e4, we have the same result except one case. We shall study, in Section 3, right almost QF rings with homogeneous socles W(k)n(Q) [7] and give certain conditions on the nilpotency m of the radical of W(k)n(Q), under which W(k)n(Q) is left almost QF or serial. In particular if m less-than-or-equal-to 2n, W(k)n(Q) is serial. We observe a special type of almost QF rings such that every indecomposable projective is uniserial or injeative in Section 4.