On primary submodules in modules of entire functions that are dual to spaces of $\Omega$-ultradifferentiable functions

We consider weighted modules of entire functions that are dual to general spaces of $\Omega$-ultradifferentiable functions. We explore the local description problem for primary submodules in these modules. It is shown that there exist non-localisable primary submodules. We also obtain non-trivial conditions under which local description is possible. All assertions may be reformulated to the equivalent dual ones concerning with the spectral synthesis problem for differentiation invariant subspaces of $\Omega$-ultradifferentiable functions.