I know this could be a dumb question, but I've been studying for hours and I might be too tired to see why:
A ring R is semisimple if and only if its global dimension is zero.
We define the global dimension of a ring as g|dim(R):=Sup{pd(M)} where M is a module finitely generated of R and pd(M) is the projective dimension of M. I already know that :
"A ring is semisimple if and only if all of its modules are projective"
which could be very useful here, but I cant see how to prove any direction of the first statement. Thanks for advance.