  Let $\mathbb{I}$ be a proper ideal of $H^{\infty}(\D)$. We prove the corona
theorem for infinitely many generators on the algebra $H^{\infty}_{\mathbb{I}}$
in which the corona theorem for finitely many functions is known to hold. This
settles the conjecture of Ryle \cite{ryle1}. We also provide the estimates for
corona solutions. Moreover, we prove a generalized Wolff's Ideal Theorem for
this sub-algebra.
