Noncoherence of some rings of functions

Let $\mathbb{D}$, $\mathbb{T}$ denote the unit disc and unit circle, respectively, in $\mathbb{C}$, with center 0. If $S\subset \mathbb{T}$, then let $A_{S}$ denote the set of complex-valued functions defined on $\mathbb{D}\cup S$ that are analytic in $\mathbb{D}$, and continuous and bounded on $\mathbb{D}\cup S$. Then $A_{S}$ 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 $A_{S}$ is not coherent.