$R$ (1 is not assumed to be in $R$) is a prime right Goldie ring (finite dimensional and ACC on right annihilators) which contains a minimal right ideal. Show that $R$ must be a simple Artinian ring.
This appeared in a past paper, the first 2 parts which I managed to prove were "Every essential right ideal of a semi-prime right Goldie ring contains a regular element" and "Every non-zero ideal of a prime ring must be essential as a right ideal". There is also an extra hint that I might have to use Artin-Wedderburn.
I've been trying (and failing) to find a way to use these results since I can't assume (or prove that) the minimal right ideal is an ideal (to use the previous result). I also have that the minimal ideal is of the form $eR$ where $e$ is idempotent, but I feel like i'm barking up the wrong tree. Any help is appreciated, thanks.