General Idea
The goal is to construct the right ring of fractions R with respect to multiplicative subset S. In other words we want to work with elements of the form as−1 and have a ring structure on the set R. The problem is that there is no obvious interpretation of the product (as−1)(bt−1); indeed, we need a method to "move" s−1 past b. This means that we need to be able to rewrite s−1b as a product b1s1−1. Suppose s−1b = b1s1−1 then multiplying on the left by s and on the right by s1, we get bs1 = sb1. Hence we see the necessity, for a given a and s, of the existence of a1 and s1 with s1 ≠ 0 and such that as1 = sa1.
Read more about this topic: Ore Condition
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