Algebras with finitely many orbits

Any algebra of finite representation type has a finite number of two-sided ideals. But there are stronger finiteness conditions that should be considered here. We consider finite-dimensional K-algebras that have only a finite number of left (respectively, principal left) ideals, up to conjugacy. We then characterize K-algebras A whose Jacobson radical satisfies J(A)2=0, and with finitely many classes of principal left ideals. Finally, we consider basic algebras with J(A)2=0. Here we characterize such algebras with finitely many classes of left ideals.