I am looking for a ring with nilpotent elements such that $J(R)=0$ where $J(R)$ is Jacobson radical.
Any suggestion?
I am looking for a ring with nilpotent elements such that $J(R)=0$ where $J(R)$ is Jacobson radical.
Any suggestion?
The best examples are the matrix rings over a field. These are simple, so they've got trivial Jacobson radical, and yet already the $2\times 2$ matrices have nilpotent elements $e_{12},e_{21}$.
There arguably is one commutative example: in the trivial ring, $1=0$ is nilpotent but the Jacobson radical is, naturally, $0.$