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Keywords:
idempotent; nilpotent; Boolean ring; local ring; Morita context
idempotent; nilpotent; Boolean ring; local ring; Morita context
Summary:
We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; $\Bbb Z_3\oplus {\Bbb Z}_3$; $\Bbb Z_3\oplus B$ where $B$ is a Boolean ring; local ring with nil Jacobson radical; $M_2(\Bbb Z_2)$ or $M_2(\Bbb Z_3)$; or the ring of a Morita context with zero pairings where the underlying rings are $\Bbb Z_2$ or $\Bbb Z_3$.
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