Finiteness and infiniteness of gradings of Noetherian rings

In this paper we show that for a torsion-free abelian group $G$, ${\mathrm{rank}}_{\mathbb{Z}}G<\mathrm{\infty }$ if and only if there exists a Noetherian $G$-graded ring $R$ such that the set $\{R_g\mathrm{\ne }0\}$ generates the group $G$. For every $G$ of finite rank, we construct a $G$-graded ring $R$ such that $R_g\mathrm{\ne }0$ for all $g\in G$. We prove such rings give examples of PIDs which are not ED. We also use the relations between the graded division ring and the group cohomology to prove some vanishing and nonvanishing results for second group cohomology. Finally, we prove that the Hilbert series of a finitely generated $G$-graded $R$-module is well-defined when $R_0$ is Artinian, and this Hilbert series times some Laurent polynomial is equal to a Laurent polynomial.