Chain conditions in special pullbacks

Let D⊆E denote an extension of commutative rings with identity, I be a nonzero proper ideal of D, Γ mean a nonzero torsion-free additive grading monoid with Γ∩-Γ={0} and Γ ^*=Γ\{0}. Let E[Γ] be the semigroup ring of Γ over E, D+E[Γ ^*]={f∈E[Γ]|f(0)∈D} and D+I[Γ ^*]={f∈D[Γ]| the coefficients of nonconstant terms of f belong to I}. In this paper, we give some conditions for the rings (resp., domains) D+E[Γ ^*] and D+I[Γ ^*] to be Noetherian (resp., to satisfy the ascending chain condition on principal ideals).