Group algebra of a p-group in characteristic p is local ($kG$ for a $p$-group $G$)

If $G$ is a finite $p$-group and $k$ has characteristic $p$, then the augmentation ideal is the Jacobson radical of $kG$ and $kG/J(kG)\cong k$. Thus $kG$ is a local ring, has only the trivial simple module, and its regular module is indecomposable.