Definition: The underlying set of the ring is countable.

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- passes to $eRe$ for any full idempotent $e$
- passes to $eRe$ for any idempotent $e$
- passes to subrings
- passes to the center
- stable under finite products
- passes to matrix rings
- Morita invariant
- passes to localizations
- passes to polynomial rings
- passes to quotient rings

- stable under products (counterexample)