Artinian rings are perfect Definition. A ring is called perfect if every flat module is projective.

Is there a simple way to prove that an Artinian ring is perfect (in the commutative case)?


The local case is proved here, Lemma 10.97.2, and then extend the result to the non-local case by using that an artinian ring is isomorphic to a finite product of artinian local rings and a module $M$ over a finite product of rings $R_1\times\cdots\times R_n$ has the form $M_1\times\cdots\times M_n$ with $M_i$ an $R_i$-module.  
