Property: finite

Definition: $R$ has only finitely many elements

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Metaproperties:

This property has the following metaproperties
  • 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
This property does not have the following metaproperties
  • passes to polynomial rings (counterexample needed)
  • passes to power series ring (counterexample needed)
  • stable under products (counterexample)