The homogeneous spectrum of a graded commutative ring

Suppose Γ is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a Γ-graded commutative ring is Noetherian if its homogeneous spectrum is Noetherian, thus answering a question of David Rush. Suppose A is a commutative ring having Noetherian spectrum. We determine conditions in order that the monoid ring A[Γ] have Noetherian spectrum. If rank Γ ≤ 2, we show that A[Γ] has Noetherian spectrum, while for each n ≥ 3 we establish existence of an example where the homogeneous spectrum of A[Γ] is not Noetherian.