Module and direct sum

Question

$R$ is a ring and $I$ is a left ideal.

If $R/I$ is an $R$ module and projective. Does this mean $R/I \oplus M \cong R$ for some $R-$module $M$? I know that $R/I \oplus M \cong R^{n}$ for some $n$ and $M$.

Answer 1

Yes. An $R$-module $P$ is projective if and only if every short exact sequence of $R$-modules $$0 \to A \to B \to P \to 0$$ splits. In your case, choose the inclusion and the projection to get an exact sequence $$0 \to I \to R \to R/I \to 0$$ in order to obtain $R = I \oplus R/I$.