localization
Localization is the universal way to legally introduce denominators. You have a ring, you choose some set of elements you wish were invertible, and localization builds the smallest ring extending yours in which exactly those elements have become units. It generalizes the leap from the integers Z to the rationals Q — there you invert everything; in general you invert only a chosen multiplicative family, getting a ring tailored to whatever you wanted to divide by.
Let R be a commutative ring and S a multiplicative subset: 1 in S and S is closed under multiplication. The localization S^{-1}R is the set of fractions a/s (a in R, s in S) under the equivalence a/s = b/t iff u(at - bs) = 0 for some u in S, with the obvious addition and multiplication. There is a canonical ring map R -> S^{-1}R, a -> a/1, and it is universal: any ring map from R sending S to units factors uniquely through S^{-1}R.
Two special cases dominate. Taking S = R \ p for a prime p gives localization at a prime, a local ring. Taking S = {1, f, f^2, ...} for a single element f gives R_f = R[1/f], the functions on the open set where f does not vanish. Localization is an exact functor on modules, commutes with quotients and finite intersections of ideals, and is flat — making it one of the gentlest, most useful operations in the subject. Caveat: localization can collapse things — if 0 lies in S the whole ring becomes the zero ring.
In Z with f = 2, the localization Z[1/2] = { a / 2^k } consists of dyadic rationals; here 2 has become a unit while odd primes remain prime, so Spec(Z[1/2]) is Spec(Z) with the point (2) removed.
Inverting 2 deletes one point from the spectrum.
When R is an integral domain and S = R \ {0}, the localization is exactly the field of fractions. The equivalence with u(at - bs) = 0 (rather than just at = bs) is what makes localization work correctly in the presence of zero-divisors.