Rings: fields of fractions

Fields of fractions

What we have done is simply to limit consideration to polynomials whose coefficients lie in a ring, namely ℤ, instead of polynomials with coefficients in a field (ℚ). Moreover, ℤ and ℚ are closely related. It turns out that many (but hardly all) other rings like ℤ can be extended to fields like ℚ in exactly the same way — ℚ is the field of fractions corresponding to the ring ℤ. Although this construction is familiar and obvious, there are subtleties about it which will motivate us to look at some interesting features of the theory of rings in general.

Let’s look at how the construction of fractions actually works. What it really comes down to is finding a solution to the simplest possible Diophantine equation: ax-b=0 for a,b∈ℤ. Doing this is something like “solving” the equation x2+1=0. Essentially, one just gives a name to the solution (x=i in the latter case) and shows that the introduction of this symbol with the property that i2=-1 does not bring along any inconsistency. One way to do this is to construct some sort of mathematical object which actually has the desired property. In the case of x2+1=0, the construction can be done in a very general way by forming the quotient ring ℝ[x]/(x2+1), noting that it is a field which contains ℝ, and has an element which satisfies the given relation. (We’ll explain this more fully a bit later.)

Now, as regards the solution of ax-b=0, we can give the name “a/b” to the solution. This a/b is constructed as follows: we start from the set of all ordered pairs of integers for which the second element isn’t 0: {(a,b)∈ℤ×ℤ | b≠0}. We want this to be a field, so there must be rules of addition and multiplication for the pairs. Multiplication is easily defined as (a,b)×(c,d)=(ac,bd). Addition is a little trickier: we define (a,b)+(c,d)=(ad+bc,bd). (That being the way one actually adds a/b and c/d.) It’s easy to check that the given set with these rules satisfies the axioms for a field. In particular, the multiplicative inverse of (a,b) is (b,a) (assuming a≠0), and the additive inverse of (a,b) is (a,-b). (One subtlety is that we have to regard (a,b) as identical to (a′,b′) in case ab′=a′b, or equivalently, to limit the original set of ordered pairs to only those where the elements have no integer divisor in common, the first element of the pair is a positive integer, and to maintain that condition by always removing common factors after addition or multiplication.)

This construction can actually be done for many, but not all, other commutative rings besides the integers ℤ. There is a certain property ℤ has which is not true of all rings. Namely, for a,b∈ℤ the product ab=0 if and only if at least one of a and b is 0. Although that is clearly true in ℤ, there are reasonable examples of rings where it fails. We’ll give such an example very soon. The property that ab≠0 whenever both factors are not 0 is so important that a commutative ring with a multiplicative identity having this property is given a special name: a domain (or sometimes, integral domain). A ring clearly needs this property to perform the construction explained in the preceding paragraph.

But that property is sufficient. The indicated construction can be done for any integral domain A, and the field that results is called the field of fractions of the domain A. Any field is also a ring, and a domain A is naturally a subring of its field of fractions by means of the correspondence a→(a,1) for all a∈A. (This is why one requires a domain A to contain a mutiplicative identity element.) A field is a domain, because if we have ab=0, and one of the factors is not 0, then multiplying by its inverse would imply that the other factor is 0. (Multiplication is always commutative in a field.)

Another standard example of a ring is a ring A[x] of polynomials over an integral domain A: A[x] consists of all finite sums of the form ∑0≤k<∞ akxk, where all ak∈A. The “x” is just a formal symbol. The rules of addition and multiplication in A[x] follow from assuming ax=xa for a∈A and applying the distributive law as necessary. Since A[x] is a domain, a field of fractions can be constructed. It consists of rational functions, which are just quotients of polynomials f(x)/g(x), where g(x)≠0.

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