Let $k$ be a field. Let $A$ be a finitely generated $k$-algebra. Consider the inclusion of the center $i:Z(A)\hookrightarrow A$. I'm interested in what general conditions there are for the pull-back functor $i^*: Z(A)\mathrm{-mod}\rightarrow A\mathrm{-mod}$ to be an equivalence of categories. Clearly, this is the identity functor if $A$ is commutative- but it seems like an interesting question to ask in the case that $A$ is non-commutative.
This seems like such an obvious question to ask that someone must have developed a theory about it- but I haven't been able to find out what the name of this theory is. I'd appreciate any answers, references, or reading suggestions that you can offer.