I had originally asked this question on math stack exchange but I think maybe it's more appropriate to ask it here.

In the paper of Beilinson, Ginzburg and Soergel entitled "Koszul Duality Patterns..." After stating the following proposition, they make the statement: " a Koszul ring is a positively graded ring that is "as close to being semisimple as it can possibly be.""

Proposition: Let $A = \bigoplus_{j\geq 0} A_j$ be a positively graded ring and suppose that $A_0$ is semisimple. The following are equivalent.

1) $A$ is Koszul

2) For any two pure $A$-modules $M$, $N$ of weights $m$, $n$ respectively we have $ext^i_A (M,N) = 0$ unless $i = m-n$.

3) $ext_A ^i (A_0, A_0 \langle n \rangle) = 0$ unless $i=n$.

To clarify some things: A graded module $M$ over a graded ring, is pure of weight $m$ iff $M = M_{-m}$. Also the $ext$'s we are using are in the category of graded modules.

I am very new to all this and I don't even have a vague idea of what they mean. Any thoughts at all would be super helpful.