Fractions

We have already got to know the fractions in our elementary school: first we get used to the operations of integers , and found none of them except for has its multiplicative inverse, such that multiplying the inverse is equivalent to dividing them; to mitigate, we introduce the concept of fractions and rational numbers , so that division is free and the operation in the original ring is compatible with the one in , and most importantly, is a field.

Following the principles of defining fractions in on , in this text, we will introduce the method of defining fractions on integral domains, and even commutative ring with identity that might have zero divisors, discussing and comparing their properties.

Axioms of fractions

Axiomatically, with the numerators and denominator taken from the commutative ring with unity , an imaginable form of fractions should support the operations of addition, multiplication and reduction shown as below:

Where are required to be non-zero. In order for them to be well-defined, the denominators should be chosen from , where is a multiplicative closed subset of . So that the numerators of the results will be closed in , while the denominators will be closed in , and the operations will be closed thereby.

Furthermore, if the fractions are guaranteed to be well-defined, then the fractions form a commutative ring with zero and unity , whose ring properties can be easily verified by:

But remember, the fractions are not naively well-defined as it looks, especially when there are zero divisors in : consider a pair of zero divisors , in the addition of fractions that , when , by the reduction rule there's , so the fraction should behave as it were zero when it encounters another fraction with denonimator . The definition of the fractions must be able to cope with the zero divisors in order to be well-defined, and the remaining text in this section is to discuss how to define and identify the fractions.

Fields of fractions

Let's begin with the relatively intuitive case, in which we are able to put aside the zero divisors. Consider the process of making into . The integers is an integral domain, and is the multiplicatively closed subset of that we choose to be the denominator. So this motivates us to try making an integral domain into fractions with being the denominator.

In the rational numbers , we identify a non-zero fraction by reducing it to reduced fraction such that . But when it comes to the integral domain , we will have to define first if we want to have a "reduced fraction", and specially has a trivial multiplicative group, but the multiplicative group might be non-trivial, and for a "reduced fraction" where , is it really more "reduced" than the fraction ? So instead of being in pursuit of a "reduced fraction", we simply identify two fractions and by interleavingly multiplying their numerators and denominators, and test whether .

So formally, we identify a fraction defined by as an equivalence class with respect to equivalence relationship :

Its being an equivalence relationship can be verified by:

1. (Reflexivity) .
2. (Symmetry) .
3. (Transitivity) .

The equivalence class for each is , and this is how we identify a fraction.

Since the fraction is an equivalence class defined for a set of equivalent elements, to ensure the operation of the fractions are well-defined, we must also ensure for every pair of element in the operand fractions, their result lies in the same equivalence class of the result fraction uniquely. For convenience, let there be two pairs of equivalent elements taken from two fractions , which means , and there will be:

1. By the axioms of fractions, the addition and multiplication are natural and closed operations defined in integral domain , so the numerators are still in . The is multiplicatively closed, so the denominators are still in and non-zeroes.
2. By the axioms of fractions, there's . Since , we have and , so the result of addition is unique.
3. By the axioms of fractions, there's . Since , we have and , so the result of multiplication is unique.
4. By , the fractions support reduction.

So the fractions defined in this way fulfil the axioms of fractions and are well-defined. And any non-zero fraction will have their numerator and denominator inside , and is thus invertible. In this way, the fractions defined by forms a field called the field of fractions of , and is denoted as or .

For example, the is an integral domain, and can be fixed into the field of fractions , which is called the field of rational functions conventionally. The same thing happens to , which is fixed into and called the field of rational functions over .

Obviously, the integral can be embedded into the field of fractions by injective ring homomorphism defined by , so there's a subring isomorphic to .

And for any field , we can say , and can be embedded into by injective ring homomorphism defined by .

To prove, in the first place, we need to prove , so that maps all equivalent elements in the same equivalence class of the fraction into the same image and is thus well-defined. By the definition of the equivalence relationship, we have . By multiplying on both sides we have . And thus we have , and is well-defined.

Then 's being a ring homomorphism can be easily proved by:

And it's obvious that , so is an injective ring homomorphism, and we are done for now.

Conversely, since any field containing will have a subfield isomorphic to , is indeed the minimum field that has embedded into it.

Specially, a field is also an integral domain, which means we can also evaluate the fields of fractions of it, but there's : first is embedded into injectively by defined by ; then by , the ring homomorphism is surjective and thus ring isomorphism.

Rings of fractions

Now let's consider the general case of commutative ring with unity , where there'll likely be zero divisors. We'll present the solution first, and then show how will the solution handle the zero divisors above.

Just like the case of field of fractions, we define the fractions by equivalence relationship that:

Its being an equivalence relationship can be verified by:

1. (Reflexivity) .
2. (Symmetry) .
3. (Transitivity) .

The equivalence class of is identified as , just as the field of fractions. And again, let's verify the uniqueness of the result in the meaning of equivalence class defined by the result fraction. Given two pairs of equivalent elements from two fractions , which means , we verify that the fractions is well-defined by:

1. Addition and multiplication are natural and closed operations defined in the ring , so the numerator are still in . If , then , so all fractions are annihilated to ; otherwise there , the denominator is still in and non-zeroes.
2. By the axioms of fractions, there's . Since , we have and , so the result of addition is unique.
3. By the axioms of fractions, there's . Since , we have and , so the result of multiplication is unique.
4. By , the fractions support reduction.

For the case of such that , noted that , all fractions whose numerator are zero divisor paired with such that .

The fractions defined in such way is called the ring of fractions of with respect to , and denoted as conventionally.

Just like the field of fractions, we have a ring homomorphism defined by , but this does not mean can be embedded in , in fact it cannot be embedded in when there exists any element such that and thus ¹.

When is not zero ring, every fraction in the form of is invertible. First if , there must be , but we have while is multiplicatively closed. So there is no such and . Then by the axioms of fractions it's clear that , so such a fraction is invertible.

If there's a ring homomorphism such that , then there's also a ring homomorphism defined by .

First, we need to show is well-defined by . By the definition of the equivalence relationship we have . By applying on both sides we have . Since is unit, it's not zero-divisor, so we have . By multiplying on both sides we have . And thus we have , and is well-defined.

Then 's being a ring homomorphism can be shown by:

Localization at prime ideals

When introducing the prime ideals , we also mention that the is multiplicatively closed, and proved that ideal 's being prime is equivalent to 's being multiplicatively closed. When building ring of fractions of with respect to (conventionally denoted as ), there're even more interesting properties.

Consider the fractions in the form of , such kind of fractions are clearly closed and form an ideal :

1. .
2. .
3. .

And for every non-zero , it's clear that and its inverse is , thus is a field. This implies is a maximal ideal of . On the other hand, every other ideal which is not a subideal of must contain an element of with , so contains a unit and thus . In this way, there's no other chain of ideals and is the only maximal ideal in . Such a rings with unique maximal ideal is called a local ring. The process utilizing prime ideal to create a local ring from into is called the localization at .

We will discuss about localization in depth when we comes to the commutative algebra, and let's just get acquainted with these concepts and move on for now.

Conclusion

In this text, we've explored and discussed two ways for defining fractions for commutative rings with unity.

For integral domain , we can just define the field of fractions with its numerator being from and its denominator being from . In this way, the ring operations of is embedded into by . Any field containing must contain a subfield isomorphic to , so is the minimum field contains .

For arbitrary commutative rings with unity, we can define the ring of fractions with its numerator being from and its denominator being from , where is one of the multiplicatively closed subset of . Although there's still ring homomorphism that , but all elements such that forms an ideal , with being the kernel of . And by first isomorphism theorem, there's , so the ring operations of cannot be embedded into under most cases. If there's a ring and a ring homomorphism such that for all element , is unit in , then there's an induced ring homomorphism defined by .

One of the most special rings of fractions is to have being the multiplicatively closed set. In this case, is a local ring with its unique maximal ideal being fractions with numerators being from and denominators being from .

The definition of fractions serves as building block with which we can compare and extend the connections between and to connections between integral domain and its field of fractions , and even to connections between commutative ring with unity and its ring of fractions , we will see them in the following texts.