Question: I had the exercise below (not graded) and I think I have all the components to solve it, but somehow I can't seem to find a proper proof. To what right $kQ$-modules do I have to relate $P(i)$ and $\mathrm{rad}(P(i))$ and how do I find the maximal ideals of the corresponding modules.
What I know: The equivalence $\mathcal{G} \colon \mathrm{Rep}_Q \to \mathrm{Mod}_{kQ}$ gives $\mathcal{G}(P(i)_j) = e_i kQ e_j$.
Define the radical of a module to be the intersection of all maximal submodules of $M$. Let $M$ be a right $A$-module. The quotient module $$\mathrm{top}(M) = M/\mathrm{rad}(M)$$ is called the top of $M$.
Exercise: Let $Q$ be a finite quiver without oriented cycles, and $P(i)$ the indecomposable right $kQ$-module associated to $i \in Q_0$.
a) We defined the radical of $P(i)$ as follows: $$ \mathrm{rad}(P(i))_j = \begin{cases} 0 & \text{if $j = i$}, \\ P(i)_j & \text{if $j \neq i$}. \end{cases} $$ Show that this definition is indeed the intersection of all maximal submodules of $P(i)$ after identifying $\mathrm{rad}(P(i))$ and $P(i)$ with the corresponding right $kQ$-modules through the equivalence $\mathrm{Rep}_Q \simeq \mathrm{Mod}_{kQ}$.
b) Determine $\text{top}(P(i))$.