Definition: The only idempotents are 0 and 1

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- passes to $eRe$ for any idempotent $e$
- passes to $eRe$ for any full idempotent $e$
- passes to subrings

- passes to quotient rings (counterexample)
- passes to matrix rings (counterexample)
- Morita invariant (counterexample)
- stable under finite products (counterexample)
- stable under products (counterexample)