Is the following true: If the Jacobson radical of a ring $R$ is semi-simple then so is the ring $R$.

Question

True or false?

$1$. If $R$ is simple as a ring then $R$ is simple as left $R$-module.

$2$. If the Jacobson radical of a ring $R$ is semi-simple then so is the ring $R$.

$1$ looks true but I'm not sure by what. $2$ looks false.

Answer 1

Actually both is false.

1. Simple rings do not need to be simple as modules over themselves. Consider for example $R=M_2(\mathbb{Q})$.

2. $J(\mathbb{Z})=0$ is semisimple, but $\mathbb{Z}$ is not semisimple.