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. Thus it is possible for R to be a right-not-left Ore domain. Intuitively, the condition that all elements of D be of the form rs−1 says that R is a "big" R-submodule of D. In fact the condition ensures RR is an essential submodule of DR. Lastly, there is even an example of a domain in a division ring which satisfies neither Ore condition (see examples below). Another natural question is: "When is a subring of a division ring right Ore?" 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 . 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 c = au = bv 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. Examples Commutative domains are automatically Ore domains, since for nonzero a and b, ab is nonzero in . 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 . Multiplicative sets The Ore condition can be generalized to other multiplicative subsets, and is presented in textbook form in and . A subset S of a ring R is called a right denominator set if it satisfies the following three conditions for every