On the flatness of the Bergman space as a module over the Hardy algebra

Let $A^2$ be the Bergman space of the open unit disc $\mathbb{D}$ in the complex plane consisting of holomorphic functions that are square integrable with respect to the area measure in the disc, and $H^{\mathrm{\infty }}$ be the Hardy algebra of bounded and holomorphic functions on $\mathbb{D}$. With pointwise operations, $A^2$ is an $H^{\mathrm{\infty }}$-module. It is shown that the $H^{\mathrm{\infty }}$-module $A^2$ is not flat. It is also shown that there exists a maximal ideal $\mathfrak{m}$ in $H^{\mathrm{\infty }}$ such that $\mathfrak{m}{A}^2=A^2$.