# Projectivity of a module

I need a hint about this question, I know all the different ways of defining a projective module, but, I don’t know where to start: R is an left Artinian ring, M is a left R-module. I need to prove M/J(R)M is a projective R/J(R) module. Could anyone please help me to prove it?

• +1 Pretty much what i would have written, except I think for completeness we might mention why $M/J(R)M$ is a $R/J(R)$ module... – rschwieb Jan 24 '18 at 12:02
• A ring R is semisimple if for every pair of $R$-modules $M\subset N$ there is a module $P$ satisfying $M\oplus P=N$. An important theorem says that the (left) semisimple rings are precisely the (left) artinian rings with trivial jacobson radical. In your case, $J(R/J)=0$. – ArtW Jan 24 '18 at 14:58