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 ⊂ C∞(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.