Q:

Showing the group ring $\mathbb ZG$ is not Artinian.

$\mathbb {Z}G $  is not Artinian where $ G$  is a finite group.
I know that $\mathbb {Z} $ is not Artinian but $\mathbb {Z} $  is not an ideal of the group ring. So How to see this? Any help would be appreciated!

A:

A slightly different way of thinking about it, which is handy in other cases too, is that

If a ring is right Artinian, all its quotients are right Artinian.

It's a corollary of left/right/two-sided ideal correspondence, of course.
For a group ring, it always happens that the coefficient ring is always isomorphic to the quotient by the augmentation ideal.  In this case, the quotient is $\mathbb Z$, clearly not Artinian.  Therefore the thing you quotiented is not Artinian either.

