On modules with d-Koszul-type submodules

The so-called weakly d-Koszul-type module is introduced and it turns out that each weakly d-Koszul-type module contains a d-Koszul-type submodule. It is proved that, M . W K . d(A) if and only if M admits a filtration of submodules: 0 . U 0 . U 1 . ... . U p = M such that all U i /U i.1 are d-Koszul-type modules, from which we obtain that the finitistic dimension conjecture holds in W K . d(A) in a special case. Let M . W K . d(A). It is proved that the Koszul dual .(M) is Noetherian, Hopfian, of finite dimension in special cases, and .(M) . gr0(E(A)). In particular, we show that M . W K . d(A) if and only if . (G(M)) . gr0(E(A)), where G is the associated graded functor.