TITLE: "as close to being semisimple as it can possibly be."
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: In the semisimple case it is really easy to calculate $ext_A^i(M,N)$, $i\ge 1$, with the above assumptions. It is zero. For Koszul rings this is almost
true, i.e., $ext^i(M,N)$ is concentrated in degree $i$ (that is, if $S$ is the direct sum of all simples, then the two natural gradings on the algebra $A^{!}=ext^{\bullet}(S,S)$ coincide). It seems reasonable that the authors had this in mind as they wrote: "Morally a Koszul ring is a graded ring that is as close to semisimple as a $\mathbb{Z}$-graded ring possibly can be”.
This was already said in the comments. I can just add another reference here, a discussion on Koszul algebras and Koszul duality, which explains several things and refers also to the paper of Beilinson, Ginzburg and Soergel, here: http://sbseminar.wordpress.com/2007/11/01/koszul-algebras-and-koszul-duality/.