Property: Boolean

Definition: For every $x\in R$, $x^2=x$.

Resources for learning about this property:

  • N. Jacobson. Basic algebra I. (2012) @ Section 8.5

Metaproperties:

This property has the following metaproperties
  • passes to $eRe$ for any full idempotent $e$
  • passes to $eRe$ for any idempotent $e$
  • passes to localizations
  • passes to quotient rings
  • passes to subrings
  • passes to the center
  • stable under finite products
  • stable under products
This property does not have the following metaproperties
  • Morita invariant (counterexample needed)
  • passes to matrix rings (counterexample needed)
  • passes to polynomial rings (counterexample needed)
  • passes to power series ring (counterexample needed)