Let $R=\mathbb F_{p}[x]/(x^{p}-1)$. $R$ has both ring and vector space structure. I am trying to show that, given a representation $\rho : \mathbb Z/ p\mathbb Z\rightarrow GL(R)$, any invariant subspace $V\subset R$ under the action of $\rho_{g}$ is indecomposable. That is, $\nexists W \subset R$ such that $R=V\oplus W$. I am also curious if I can write down all the invariant subspaces $V_{i}$ under $\rho$.
Here is what I have done so far:
I proved any intertwining map $T: R \rightarrow R$ is given by multiplication by some $r\in R$. I also proved that $(x-1)^p=x^p-1=0$ which implies that the minimal polynomial does not have distinct roots in $\mathbb F_{p}$. I am hoping to deduce that $u^2=u\space \forall u\ne 0,1$ in $R$ in order to show no nontrivial projection $R\rightarrow V$ exists, since intertwining maps are given by multiplication by elements of $R$. Hence, there is no decomposition.