Example of a faithful irreducible module

Question

Let $R$ be a non trivial simple ring. I am trying to show that there is a faithful irreducible left $R$-module.

Is the ring $R$ considered as a left module over itself such a module? I think it's faithful since the ring map $R \to R_\ell: r\mapsto (r_\ell:a\mapsto ra))$ has zero kernel but I am not sure that it's irreducible. How do you show that it has no proper left ideals?

Many thanks.

Answer 1

Over a simple ring $R$, every nonzero unital module is faithful.

So just take any simple $R$ module $S$ and you have an irreducible faithful module.