In the paper 'Koszul Duality Patterns in Representation Theory' by Beilinson et. al, they give the definition of a Koszul Ring:
A Koszul ring is a positively graded ring $A = \bigoplus_{j \geq 0} A_j$ such that (1) $A_0$ is semisimple and (2) $A_0$ considered as a graded left $A-$module admits a graded projective resolution:
$\cdots \rightarrow P^2 \rightarrow P^1 \rightarrow P^0 \twoheadrightarrow A_0$
such that $P^i$ is generated by its degree $i$ component, i.e., $P^i = AP^i_i$.
My question is: How is $A_0$ a graded left $A-$module? Is a module with $A_0$ in degree zero, all higher graded components equal to zero, where all higher graded elements of $A$ act by zero? (In other words, $A_0$ is not a submodule of $A$, but rather a quotient module $A_0 = A/A_{>0}$).
And: What does it mean for $A_0$ to be semisimple? Semisimple as a left $A-$module? (Which would be the same as being semisimple as a left module over itself?).
Thanks.
