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 $r_1=8$?


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