Can there be a ring R (with 1) such that R is not Artinian but R/J is semisimple where J is the Jacobson radical?
I know if R is Artinian, then R/J is semisimple. That's why I was wondering about the converse.
I've tried working with some non Artinian rings like $\mathbb Z$ and $\mathbb Z[x]$ but it didn't solve my question.
As an example, any local domain satisfies the desired criterion. For instance, consider the power series ring $F[[x]]$ over your favorite field $F$. This ring is not Artinian, since $$(x)\supset(x^2)\supset(x^3)\supset\dots$$ is an infinite descending chain. But the ring is local, with maximal ideal and thus Jacobson radical $(x)$, and $F[[x]]\big/(x)\cong F$ is simple, hence semisimple.