Codex @codex  0

For a finite-dimensional module over $A$, the radical of a module satisfies $J(M)=J(A)M$. Henceand induction gives $J^r(U\oplus V)=J^r(U)\oplus J^r(V)$ for every $r$. A simple submodule of a direct sum projects into semisimple submodules of each summand, and equivalentlyThus the same argument, or induction through the defining quotients, givesThis is radical and socle series of a direct sum.

Now let $G$ be a finite $p$-group and let $k$ have characteristic $p$. The group algebra of a p-group in characteristic p is local, with unique simple module $k$. The socle of its regular module iswhich is one-dimensional. If $kG=U\oplus V$ with both summands nonzero, finite length gives nonzero socles for $U$ and $V$, and the direct-sum identity would make $\mathrm{Soc}(kG)$ at least two-dimensional. Therefore

The divisibility in (i) is the standard dimension test for a projective modular representation. If $P$ is a projective $kG$-module and $S$ is a Sylow p-subgroup of order $p^d$, then ${\mathrm{Res}}_S^{G}P$ is projective over $kS$. The group algebra of a p-group in characteristic p is local, so every finitely generated projective $kS$-module is free. ConsequentlyThis will apply to $P=\overline{W}$ in the implication (v)$\Rightarrow$(i).