One thing we haven’t given good examples of is fields. We can
get some from factoring out a maximal ideal from a commutative
ring with unit, but the most familiar example — rational numbers
— comes from a different construction. First we define a
multiplicatively closed set. This is a subset of a commutative
ring with unit which is, predictably enough, closed under the
ring multiplication. We also require for technicality’s sake
that contains the unit . A good place to get such
multiplicatively closed sets is as complements of prime ideals —
given two elements and in but not in the prime ideal , their
product must also be outside . Another good way is to start with
some collection of elements and take the submonoid they generate
under multiplication. In general not all the elements of will be
invertible in . What we want to do is make a bigger ring that
properly contains (a homomorphic image of) in which all elements
of do have inverses. We’ll do this sort of like how we built the
integers by adding negatives