I search for a ring $R$ with Jacobson radical $J(R)$ such that $R/J(R)$ is not semisimple Artinian.
Being a finitely generated module over itself, $R$ would have infinite hollow dimension due to a fact stated by Sarath and Varadarajan to the effect that an $R$-module $M$ has finite hollow dimension only if $M/J(M)$ is a semisimple Artinian module, where $J(M)$ is the Jacobson radical of $M$. Thanks in advance.
For any field $F$, $R=\prod_{i=1}^\infty F$.
It has $J(R)=\{0\}$ (since it is von Neumann regular) and is clearly not Artinian.
Then if you want a version with a nonzero radical, you can take the 2 by 2 upper triangular matrix ring over this ring. Another example would be to take the subring of this matrix ring of elements with constant diagonal.