Closed Submodules of Holomorphic Functions with Two Generators

Let $I$ be a closed submodule over a polynomial ring in a space of holomorphic functions on a domain in the complex plane. We establish sufficient conditions under which $I$ is generated by two functions or two special submodules. As a corollary, it follows from these results that if an invariant subspace $W\subset C^{\infty}(a,b)$ (with respect to the differentiation operator) admits spectral synthesis, then it is the solution space of a system of two homogeneous convolution equations.