Corollary

The submodule of E generated by $e$ is the vector subspace generated by the $e_n$.

This follows from the formulas (2).

The integers $n\geq 0$ such that $e_n\not= 0$ constitute an interval in $\mathbf{N}$, and the corresponding elements $e_n$ form a basis over $k$ of the submodule generated by $e$ (indeed, they are linearly independent because they are non-zero elements of distinct weights). This basis will be said to be associated to the primitive element $e$.