On the coherence of some irrational transfer function classes

Let D, T denote the unit disc and unit circle, respectively, in C, with center 0. If S T, then let AS denote the set of complex-valued functions dened on D[S that are analytic in D, and continuous and bounded on D [ S. Then AS is a ring with pointwise addition and multiplication. We prove that if the intersection of S with the set of limit points of S is not empty, then the ring AS is not coherent.