Let R = ⊕n∈0 Rn be a positively graded Noetherian commutative ring. Set R+: = ⊕n∈Rn. Let N = ⊕n∈ Nn be a nonzero finitely generated graded R-module. Here, 0, , and denote the set of non-negative, positive, and all integers, respectively. Let (P) denote the properties: (i) for all i ∈ 0 and all n ∈ , the R0-module [image omitted] is finitely generated; (ii) for all i ∈ 0, [image omitted], where end(T) = Sup{r | Tr ≠ 0}. In this note, we study the following question: For a graded ideal I contained in R+, when does [image omitted] have the properties (P)? Further, we study the tameness of [image omitted].