A module $M_R$ is called uniserial if the set of all of its submodules is a chain with respect to the inclusion (e.g., the $\mathbb{Z}$-module $\mathbb{Z}_{p^m}$, for every positive integer $m$, are uniserial). $M_R$ is called serial if it is the direct sum of uniserial modules. Indeed, an indecomposable module is serial if and only if it is uniserial. A ring $R$ is called right (resp. left) uniserial if the module $R_R$ (resp. ${}_RR$) is uniserial, and $R$ is right (resp. left) serial if the module $R_R$ (resp. ${}_RR$) is serial. A ring $R$ is serial if it is both left and right serial. For instance, the ring $\mathbb{Z}_n$ for every $n > 1$ is serial.
An example of a finite artinian (left and right artinian) serial ring $R$ with $J^2=J^2(R)=0$ is $\mathbb{Z}_{pq}$ where $p,q$ are positive primes (not necessarily distinct).
I wonder if there exists an example of an infinite artinian serial ring with $J^2=0$ which is not semisimple?!. I appreciate if the details of the example are included.
Thanks in advance.
How about $\mathbb R[x]/(x^2)$? It's infinite, Artinian, uniserial, and has a nonzero radical $(x+(x^2))$ squaring to zero.
Or $T_2(\mathbb R)$ (upper triangular matrices), whose radical is $\begin{bmatrix}0&\mathbb R \\ 0& 0\end{bmatrix}$ squaring to zero.
You didn't specify if $J(R)$ had to be nonzero, and in that case any semisimple ring would work.