Show $\mathbb{Z}G$ is a noetherian ring for finite group $G$

Question

Let $G$ be a finite group. Show that $\mathbb{Z}G$ is noetherian as a ring.

It's noetherian as a $\mathbb{Z}$-module (this follows because $\mathbb{Z}$ is noetherian and $\mathbb{Z}G$ isomorphic to $\mathbb{Z}^{|G|}$ as a $\mathbb{Z}$-module). From here it 'looks' noetherian because (speaking very very loosely) it seems like once you account for the ascending chain condition of $\mathbb{Z}$, you're only going to get finitely many new ideals by introducing multiplication by linear combinations of elements of $G$... (I know... very imprecise... but it's only nebulous intuition at this stage).

The question is from a first course in module theory. Relevant level might be similar to the presentation in Jacobson's Basic Algebra.

Answer 1

If it is a Noetherian $Z$-module, consider an ideal of $ZG$, it is a $Z$-submodule of $ZG$, thus it is finitely generated over $Z$, this implies that it is finitely generated ring.

Answer 2

Hint: show that being finitely generated over $\Bbb Z$ implies finitely generated over $\Bbb ZG$.