Examples
Commutative domains are automatically Ore domains, since for nonzero a and b, ab is nonzero in aR∩bR. Right Noetherian domains, such as right principal ideal domains, are also known to be right Ore domains. Even more generally, Alfred Goldie proved that a domain R is right Ore if and only if RR has finite uniform dimension. It is also true that right Bézout domains are right Ore.
A subdomain of a division ring which is not right or left Ore: If F is any field, and is the free monoid on two symbols x and y, then the monoid ring does not satisfy any Ore condition, but it is a free ideal ring and thus indeed a subring of a division ring, by (Cohn 1995, Cor 4.5.9).
Read more about this topic: Ore Condition
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