Let $I$ ba a two-sided ideal of a ring $R$ and $M$ be an $R-$ left module. Prove that if $M$ is an $R-$ projective left module then $M/IM$ is an $R/I-$ projective left module.
It is easy to see that $M/IM$ has the structure of $R/I-$module. I need some ideas to prove that $M/IM$ is an $R/I-$ projective left module.
Thank you very much.
Show that $\hom_{R/I}(M/IM,-) \cong \hom_R(M,U(-))$, where $U$ is the forgetful functor from $R/I$-modules to $R$-modules. Hence, this is a composition of two exact functors, hence exact.