The homogeneous spectrum of a graded commutative ring

Suppose $\Gamma$ is a torsion-free cancellative commutative monoid for which the group of quotients is finitely generated. We prove that the spectrum of a $\Gamma$-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 $A[\Gamma]$ have Noetherian spectrum. If $\rank \Gamma \le 2$, we show that $A[\Gamma]$ has Noetherian spectrum, while for each $n \ge 3$ we establish existence of an example where the homogeneous spectrum of $A[\Gamma]$ is not Noetherian.