A Proof Of Ore's Theorem

The classical construction of the rational numbers involves consideration of certain equivalence classes of ordered pairs [(a,b)] where a and b are integers with b nonzero. An elementary generalization of this idea is Ore's Theorem which gives a necessary and sufficient condition that a ring, not necessarily commutative and not necessarily a domain of integrity, can be extended to a ring of "fractions." The purpose of this thesis is to analyze another proof of Ore's Theorem which involves a bare minimum of technique using the method of maximal extensions of semi-endomorphisms defined on a certain class of right ideals, i.e., given a ring with Ore's Condition we will construct the classical ring of right quotients.