Non-Artinian algebras over which proper cyclic right modules are Artinian must be right Ore domains. It is shown that if R is a PI-ring whose proper cyclic right R-modules are Artinian, then R is right Noetherian. In particular, if G is a solvable group and each proper cyclic right K[G]-module is Artinian, then the group algebra K[G] is Noetherian. It is also shown that for a group algebra K[G], if every proper cyclic right K[G]-module is Artinian and K-finite dimensional, then K[G] is Noetherian.