The strong no loop conjecture for mild algebras

Let Λ be a finite dimensional associative algebra over an algebraically closed field with a simple module S of finite projective dimension. The strong no loop conjecture says that this implies , i.e. that the quiver of Λ has no loop at the point corresponding to S. In this paper we prove the conjecture in case Λ is mild, which means that Λ has a distributive lattice of two-sided ideals and each proper factor algebra Λ/J is representation-finite. In fact, it is sufficient that a “small neighborhood” of the support of the projective cover of S is mild.