Question about simple module over semisimple Artinian ring

Question

Let $R$ be a semisimple Artinian ring and $M$ be a simple left $R$-module. I need to show that there exists some left ideal $I$ of $R$ such that $M \oplus I \cong R$. I am not really sure how to do this. I know that any module over a semisimple Artinian ring is projective, meaning it is a summand of a free module.

Answer 1

Since $M$ is simple, it is generated by a single element, so there exists some module epimorphism $f:R\rightarrow M$. (Let $m$ be a generator of $M$, and define $f(r) = rm$ for $r \in R$.) Since $M$ is projective, this epimorphism splits, so $R \cong M \oplus\operatorname{Ker}f$.