I apologize in advance for my vague question.
For an algebra course I am taking at the moment we have to write a short paper about a chosen subject, I chose to prove the Morita equivalence of finite dimensional $K$-algebra with a bound quiver algebra (where $K$ is algebraically closed).
My question is the following, what information does the category of (for example) left-modules over a ring contain? The reason for me asking is that Morita equivalence is of course coarses than isomorfism, so obviously not all information is contained in the category of modules, though what is still?
I have read somewhere (https://qchu.wordpress.com/2012/02/06/centers-2-categories-and-the-eckmann-hilton-argument/) that for a commutative ring this contains all information, this was a consequence that one could reconstruct the centre from the information contained in the category of modules. So I suppose for a commutative ring $R$, to 'understand' this ring it is enough to study its associated $R$-module category?
What about non commutative rings?
Any help would be greatly appreciated.