= Solution
An $A$-module is a <semisimple module> when it is a direct sum of <simple modules>, equivalently when every submodule has a complementary submodule. The finite-dimensional algebra $A$ is a <semisimple algebra> when its left regular module ${}_AA$ is semisimple.
Let $C_{p^n}=\langle g\rangle$, put $N=p^n$, and suppose $\operatorname{char}k=p$. The <freshman's dream> gives
$$
g^N-1=(g-1)^N,
$$
so, with $u=g-1$,
$$
kC_{p^n}\cong k[u]/(u^N).
$$
The <indecomposable modules of a cyclic p-group in characteristic p> are precisely
$$
\boxed{M_r=k[u]/(u^r),\qquad1\leq r\leq N.}
$$
Indeed, a module is a vector space with a nilpotent operator $u$, and its decomposition into <Nilpotent Jordan blocks> gives these modules. Each $M_r$ is a <uniserial module>, with unique chain
$$
M_r\supset uM_r\supset\cdots\supset u^{r-1}M_r\supset0.
$$
Therefore
$$
\boxed{J(M_r)=uM_r,\qquad\operatorname{Soc}(M_r)=u^{r-1}M_r.}
$$
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