as any good ta knows, adding fractions can be tricky. but being able to work with fractions can enrich your mathematical life in many ways. unfortunately, if you happen to be a ring, this luxury is not inherent. however, you could still hope to embed into some larger ring containing inverses--even to the extent that you become a field or division ring. we will look first at the theory of localization for commutative rings where everything is fairly straightforward. then, we will see what happens when we try to generalize to noncommutative rings and look at some examples of good and bad cases. and the quote... thoreau.