Definition: $R$ has only finitely many elements

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

- passes to polynomial rings (counterexample needed)
- passes to power series ring (counterexample needed)
- stable under products (counterexample)