On Primary Submodules in Modules of Entire Functions That Are Dual to Spaces of Ω-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 primary submodules that are not weakly localizable. Nontrivial conditions are obtained under which a local description is possible. All assertions can be reformulated as equivalent dual ones concerning the spectral synthesis problem for differentiation-invariant subspaces of $$\Omega $$ -ultradifferentiable functions.