Let $R$ be a non-commutative Ore domain, and let $R[S_i^{-1}]$ be a finite set of Ore localizations. The absence of a geometric theory means there is no obvious candidate for when this set of localizations is a `cover' (correct me if I am wrong on this). A weaker requirement is that $$ \bigcap_i R[S_i^{-1}] = R,$$ where the intersection is in the skew-field of fractions of $R$.

**Question:** Assume that $R$ is an intersection of finitely-many Ore localizations $R[S_i^{-1}]$. What ring-theoretic properties of $R$ can be checked on the $R[S_i^{-1}]$?

That is, I am looking for properties $P$ such that, if each $R[S_i^{-1}]$ has property $P$, then $R$ also has property $P$.

Examples of properties I have in mind:

- Finitely-generated
- Noetherian
- Cohen-Macaulay

Properties I can show can be checked on the localizations:

- There are no nontrivial module-finite extension rings contained in the skew-field of fractions. (I don't know the name of this, but in the commutative case, it is one definition of an
**integrally-closed**domain).

`$0 \to R \to \bigoplus R[S_i^{-1}] \to \bigoplus R[S_i^{-1}, S_j^{-1}] \to \bigoplus R[S_i^{-1}, S_j^{-1}, S_k^{-1}] \to \cdots$`

be exact. – David Speyer Mar 8 '12 at 23:05