ALMOST QF RINGS WITH J(3)

In this paper we always assume that R is a two-sided artinian ring wit h identity. In [3] we have defined right almost QF rings and showed th at those rings coincided with rings satisfying ()* in [2], which K. O shiro [5] called co-H rings. We shall show in Section 2 that right alm ost 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-side d 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 orthogona l primitive idempotents. Moreover if 1 = e1 + e2 + e3 + e4, we have th e same result except one case. We shall study, in Section 3, right alm ost QF rings with homogeneous socles W(k)n(Q) [7] and give certain con ditions 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-eq ual-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 injeat ive in Section 4.