Having some non-zero finite ideal $I$ of prime ring with identity $R$ prove that $R$ is also finite.
My only guess would be to show that if $R$ was infinite, only a finite number of its elements would be non-zero (basing on the definition of a prime ring). However, I'm struggling to find something to start with. I would be greatful for a direction of thought or a reference to a similar task.
Since $I$ has left annihilator zero, the map $R\to End(I_R)$ which sends $r$ to the endomorphism of $I$ given by left multiplication by $r$ is an injective ring homomorphism. But $End(I_R)$ is of course finite since $I$ is.
We can continue to conclude that $R$ is simple (any Artinian prime ring is) and $I=R$, and that $R$ is a matrix ring over a finite field.