A right Artinian ring has finite representation type when it has only finitely many isomorphism classes of finitely generated indecomposable modules on the right. For such rings, this is equivalent to right pure semisimplicity together with finiteness of the set of indecomposable right-module isomorphism classes (Prest 1988).
The pure semisimplicity conjecture asks whether the latter finiteness follows from right pure semisimplicity alone. The Artinian or pure-semisimple hypotheses matter when transferring these characterizations to general rings.
