Finite generation of Noetherian graded rings

Abstract:

Let $H$ be an additive abelian group. Then a commutative ring $A$ is said to be $H$-graded if there is given a family ${\{ {A_h}\} _{h \in H}}$ of subgroups of $A$ such that $A = { \oplus _{h \in H}}{A_h}$ and ${A_h}{A_g} \subset {A_{h + g}}$ for all $h$, $g \in H$. In this note it is proved, provided $H$ is finitely generated, that an $H$-graded ring $A$ is Noetherian if and only if the ring ${A_0}$ is Noetherian and the ${A_0}$-algebra $A$ is finitely generated. This is not true in general unless $H$ is finitely generated. A counterexample is given.