Reference(s):

• T. Lam. A first course in noncommutative rings. (2013) @ Chapter 5

Metaproperties:

This property has the following metaproperties

• passes to $eRe$ for any full idempotent $e$
• passes to the center
• passes to quotient rings

This property does not have the following metaproperties

• Morita invariant (Counterexample: $R_{ 12 }$)
• passes to matrix rings (Counterexample: $R_{ 12 }$)
• passes to $eRe$ for any idempotent $e$ (counterexample needed)
• passes to polynomial rings (Counterexample: $R_{ 7 }$ is the polynomial ring of $R_{ 2 }$)
• passes to power series ring (Counterexample: $R_{ 27 }$)
• stable under finite products (Counterexample: $R_{ 9 }$)
• stable under products (Counterexample: $R_{ 57 }$)
• passes to subrings (Counterexample: $R_{ 6 }$ is a subring of $R_{ 101 }$)
• forms an equational class (counterexample needed)