Finitely generated submodules of differentiable functions II

Let $\text{E}$(Ω)]^p be the Cartesian product of the space of real-valued infinitely differentiable functions on a connected open set Ω in $\text{R}$ⁿ with itself p-times. The finitely generated submodules of $\text{E}$(Ω)]^p are of the form im(F) where F: $\text{E}$(Ω)]^q → $\text{E}$(Ω)]^p is a p × q matrix of infinitely differentiable functions on Ω. Let $\text{r =}\text{max}\text{{}\text{rank}\text{(F(x)): x ? Ω}}$. The main results of the present paper are that for Ω ? $\text{R}$ⁿ, if the finitely generated submodule im(F) is closed in $\text{E}$(Ω)]^p, then for every x?ω with rank(F(x)) < r there exists an r × r sub-matrix A of F such that x is a zero of finite order of det(A), and for Ω ? $\text{R}$¹ the converse also holds.