I'm looking for an example of a ring $R$ (necessarily nonunital) which is simple (in the sense that $R \cdot R \neq 0$ and $R$ has no proper, nonzero 2-sided ideals) and also radical (in the sense that the Jacobson radical $J(R)$ is all of $R$). My only thought so far has been that it suffices to find a simple ring in which all elements are nilpotent. Thanks.
The first example of a simple radical ring was constructed in 1961 by E. Sasiada, see:
The problem of the existence of a simple nil ring remained open until 2002, when it was solved by A. Smoktunowicz, see:
I did some googling and found the following book in which Sasiada's construction is presented in the form of an exercise.
Added: I did some further googling and found the following article which might also be of interest.