# What information about a ring does the category of modules of this given ring contain?

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.

• There is no “canonical” element in the Morita equivalence class of a ring, in general. However a commutative ring or a basic semiperfect ring in this class is unique (up to isomorphism). – egreg Jun 7 '17 at 17:32
• – Torsten Schoeneberg Nov 23 '17 at 19:57

## 1 Answer

There are lots of module-category properties that translate to some ring property. Namely,

1. $R$ is right Noetherian if all f.g. $R$ modules are Noetherian (similarly with Artinian)
2. All $R$ modules are projective (or all modules are injective) $\iff$ $R$ is semisimple
3. All $R$ modules are flat (or divisible, in a certain special sense given by Lam) $\iff$ $R$ is von Neumann regular
4. A ring $R$ is left perfect $\iff$
1. All left modules have projective covers $\iff$
2. Left projective and left flat modules coincide
5. The injective modules coincide with projective modules iff $R$ is quasi-Frobenius
6. A ring is left hereditary (all left ideals are projective) $\iff$ submodules of projective left modules are all projective.

There are even a lot of circumstances where ring theoretic properties can be characterized by f.g., or cyclic, or simple modules alone. For information on that I'd recommend looking at this book on the subject

Typically if there is a nice module category property, we come up with a special name for the rings that have that property on their modules, so it is pretty difficult to think of a module category property that doesn't have a ring theoretic property associated with it (and if you did think of one, you could immediately coin a class of rings to match with it.)