Rings and Semi-simple rings

Question

I'm failing to see which of the following are semi-simple rings, any help would be appreciated.

$\mathbb{C}[X]$, the group ring $\mathbb{Q[Z]}$ and $\begin{pmatrix} \mathbb{Z} & \mathbb{Q}\\ 0 & \mathbb{Q} \end{pmatrix}$

Thanks.

Answer 1

Hints:

$\Bbb C[X]$ is a domain, and a domain is Artinian only if it's a field...

$\Bbb Q[\Bbb Z]$ is a group algebra over an infinite group, but a group algebra over a field is Artinian iff the group is finite...

The last ring $\begin{pmatrix} \mathbb{Z} & \mathbb{Q}\\ 0 & \mathbb{Q} \end{pmatrix}$ has a nonzero nilpotent ideal, hence its Jacobson radical is not zero. Find the nilpotent ideal. Alternatively, you can work to show it's not Artinian. You can, in fact, find a two-sided ideal such that the quotient is $\Bbb Z$. Can you see why this implies the ring isn't Artinian?