# Finite representation-infinite rings

Rings are not necessarily commutative, but associate and unital here. Recall that representation-infinite means that there are infinite non-isomorphic indecomposable modules. For a natural number $$m$$ define

$$r_m:= \inf \{ n \geq 1 |$$ there exists a representation-infinite finite connected ring with $$n$$ elements and $$m$$ simple modules $$\}$$ .

Question: What is the sequence $$r_m$$?

I think we have $$r_1=8$$, which means that the smallest representation-infinite finite connected ring has 8 elements. It should be $$K[x,y]/(x^2,y^2,xy)$$, when $$K$$ is the field with 2 elements.

Question: Is there an elementary argument for $$r_1=8$$?