Definition: $R$ is a commutative domain with field of fractions $K$, and $K=R[u^{-1}]$ for some nonzero $u\in R$. Equivalently: $K$ is finitely generated as a ring over $R$; Equivalently: There is a nonzero $u\in R$ contained in all nonzero prime ideals of $R$

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- passes to localizations

- passes to polynomial rings (Counterexample: $R_{ 6 }$)
- passes to quotient rings (Counterexample: $R_{ 49 }$ is a homomorphic image of $R_{ 27 }$)
- passes to subrings (Counterexample: $R_{ 6 }$ is a subring of $R_{ 101 }$)
- stable under finite products (Counterexample: $R_{ 9 }$)
- stable under products (counterexample needed)
- forms an equational class (counterexample needed)

Rings

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- = has the property
- = does not have the property
- = information not in database