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I am looking for a ring with nilpotent elements such that $J(R)=0$ where $J(R)$ is Jacobson radical.

Any suggestion?

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    https://en.wikipedia.org/wiki/Semiprimitive_ring It seems Kevin's example below is von Neumann regular.2012-08-07

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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, $0.$

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    I often wonder whether having a terminal object in Rings is really worth all the trouble.2012-08-07