QF - 1 Rings of Global Dimension ≦ 2

Abstract

R. M. Thrall [10] introduced QF — 1, QF — 2 and QF — 3 rings as generalizations of quasi-Frobenius rings. (For definitions, see section 1. It should be noted that all rings considered are assumed to be left and right artinian.) He proved that QF — 2 rings are QF — 3 and asked whether all QF — 1 rings are QF — 2, or, at least, QF — 3. In [9] we have shown that QF — 1 rings are very similar to QF — 3 rings. On the other hand, K. Morita [6] gave two examples of QF — 1 rings, one of them not QF — 2 and therefore not QF — 3, the other one QF — 3, but not QF — 2.