Let $R$ be a ring with identity. The ring $R$ is semisimple if it is semisimple as a left $R$ module. A module $M$ is semisimple if it can be expressed as a direct sum of simple submodules. The Jacobson radical of $R$, denoted by $J(R)$, is the intersection of all maximal left ideals of $R$.
I am asked to prove this:
The ring $R$ is semisimple if and only if it is Artinian and $J(R) = 0$.
I have proved the $\implies$ part. How to prove the converse? Since $R$ is Artinian every collection of left ideals of $R$ has a minimal element. How to go from here?