A ring of symmetric Hermitian modular forms of degree $2$ with integral Fourier coefficients

We determine the structure over $\mathbb{Z}$ of the ring of symmetric Hermitian modular forms with respect to $\mathbb{Q}(\sqrt{-1})$ of degree $2$ (with a character), whose Fourier coefficients are integers. Namely, we give a set of generators consisting of $24$ modular forms. As an application of our structure theorem, we give the Sturm bounds of such the modular forms of weight $k$ with $4\mid k$, in the case $p=2$, $3$. We remark that the bounds for $p\ge 5$ are already known.