Local rings with simple radical

Question

Is there a (finite) non-commutative local ring $R$ (containing identity) such that $J(R)$ is simple as a left module?

Answer 1

For a finite field $k$ with non-trivial automorphism $\sigma$, take the skew polynomial ring $k[X, \sigma]$ (reminder: these are polynomials with coefficients on the left $\sum a_i X^i$ with relation $Xa = \sigma(a)X$) and set $R := k[X, \sigma] / \langle X^2 \rangle$ (i.e. divide out the (two-sided) ideal generated by $X^2$).
$R$ is local with maximal ideal $J(R) = \langle X \rangle$ which as a left $R$-module is isomorphic to $R/J(R) = k$, in particular it is simple.