question about Deligne and Husemöller's "Survey of Drinfel'd modules"

Question

I am trying to read Deligne and Husemöller's notes on Drinfel'd modules. They consider a situation where $\phi: A \rightarrow \mathrm{End}_{k}(G_{a})$ is an injective ring homomorphism from a Dedekind domain $A$ into the ring of $k$-endomorphisms of the group scheme $G_{a}$ for some field $k$ of positive characteristic. This induces a functor $E$ from the category of $k$-algebras into the category of $A$-modules. Given any $a \in A$, the functor $E$ has a subfunctor $E_{a}$, where $E_{a}(R)$ is the submodule of $E(R)$ annihilated by $a$ for any $k$-algebra $R$. Then, letting $\overline{k}$ be an algebraic closure of $k$, they consider the $A/(a)$-module $E_{a}(\overline{k})$ for some $a \in A$ such that $a \neq 0$ and speak about its ``primary components". So perhaps they are implying that $E_{a}(\overline{k})$ has a reduced primary decompomsition when viewed as a submodule of $E(\overline{k})$. But the module $E(\overline{k})$ would seem not to be finitely generated, so I find myself stuck about how to prove that the submodule $E_{a}(\overline{k})$ would have a reduced primary decomposition.