It is important to remember that the definition of R being a right order in D includes the condition that D must consist entirely of elements of the form rs−1.
Any domain satisfying one of the Ore conditions can be considered a subring of a division ring, however this does not automatically mean R is a left order in D, since it is possible D has an element which is not of the form s−1r.
One characterization is that a subring R of a division ring D is a right Ore domain if and only if D is a flat left R-module (Lam 2007, Ex.
A different, stronger version of the Ore conditions is usually given for the case where R is not a domain, namely that there should be a common multiple with u, v not zero divisors.
In this case, Ore's theorem guarantees the existence of an over-ring called the (right or left) classical ring of quotients.
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