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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?

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Since R/J is semisimple artinian, all short exact sequences of left R/J-modules split, so in particular all left R/J-modules are projective.

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    $\begingroup$ +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... $\endgroup$ – rschwieb Jan 24 '18 at 12:02
  • $\begingroup$ I agree - but as the OP asked for a hint so I might have written too much $\endgroup$ – ArtW Jan 24 '18 at 12:06
  • $\begingroup$ @ArtW since I don’t know anything about semisimple yet, Is there any other suggestion? $\endgroup$ – Math90 Jan 24 '18 at 14:24
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    $\begingroup$ 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, $J(R/J)=0$. $\endgroup$ – ArtW Jan 24 '18 at 14:58

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