Question

Let \(R\) be a UFD, \(K\) the quotient field of \(R\). Show that every element \(z\) of \(K\) may be written \(z=a / b\), where \(a, b \in R\) have no common factors; this representative is unique up to units of \(R\).

Short answer

In summary, we have shown that every element of the quotient field \(K\) of a Unique Factorization Domain (UFD) \(R\) can be written as a fraction \(z = a / b\), with \(a\) and \(b\) belonging to \(R\) and having no common factors. Moreover, we have proved that this representation is unique up to the units of \(R\). This result is significant because it allows us to study the elements of the quotient field in a unique and simple representation while considering elements from the original domain \(R\).

Step-by-step solution

Existence of the desired representation

Let \(z\) be an element of \(K\), the quotient field of \(R\). By definition of a quotient field, we can write \(z=a / b\) for some \(a,b \in R\), \(b \neq 0\). To show that a representation with no common factors exists, we can consider the greatest common divisor (GCD) of \(a\) and \(b\). As \(R\) is a UFD, we are allowed to take GCD. Let \(d = gcd(a, b)\). Since \(d\) divides both \(a\) and \(b\), we can write \(a = d\cdot a'\) and \(b = d\cdot b'\) for some \(a', b' \in R\). We can now define a new fraction using \(a'\) and \(b'\), say \(z' = \frac{a'}{b'}\). Notice that $$ z = \frac{a}{b} = \frac{d a'}{d b'} = \frac{a'}{b'}, $$ since we can cancel the common factors \(d\). Now, \(gcd(a', b') = 1\). Thus, we have found the desired representation of \(z\) with no common factors.

Uniqueness up to units of \(R\)

To prove uniqueness, suppose we have two different representations of \(z\) as fractions with no common factors: $$ z = \frac{a_1}{b_1} = \frac{a_2}{b_2}, $$ where \(gcd(a_1, b_1) = gcd(a_2, b_2) = 1\). We want to show that \(a_1 = u\cdot a_2\) and \(b_1 = u\cdot b_2\) for some unit \(u \in R\). Cross-multiplying, we obtain: $$ a_1 b_2 = a_2 b_1. $$ Now, since \(R\) is a UFD, every element can be factored uniquely up to units, which means if prime factors of some element \(c\) also divide both \(a\) and \(b\), then they must divide \(gcd(a,b)\). Since \(gcd(a_1, b_1) = gcd(a_2, b_2) = 1\), we know that no prime factors of \(a_1\) can divide \(b_1\), and similarly, no prime factors of \(a_2\) can divide \(b_2\). However, from the previous equation, prime factors of \(a_1\) must also be prime factors of \(a_2 b_1\). Now we can deduce that the prime factors of \(a_1\) divide \(a_2\). Let these prime factors be \(\{p_1, p_2, ..., p_n\}\). Therefore, we can say that \(a_2 = u \cdot a_1\) for some unit \(u \in R\). Now, substituting this back to our equation, we get $$ a_1 b_2 = u \cdot a_1 b_1, $$ now if \(a_1 \neq 0\), we can cancel out \(a_1\) which yields \(b_2 = u\cdot b_1\) just as we wanted. If \(a_1 = 0\), we have \(z=0\) and the equation becomes: \(0 = u\cdot 0\), which also satisfies the condition of the uniqueness up to units of \(R\). This completes the proof.

Key concepts

Quotient Field

In algebra, a quotient field is the field of fractions formed from an integral domain. An integral domain is a ring \(R\) where no two non-zero elements multiply to zero. The quotient field, often denoted as \(K\), includes all numbers that can be expressed as fractions \(a/b\) where \(a, b \in R\) and \(b eq 0\).

In practical terms:

• Any non-zero element in \(R\) acts as a potential denominator.
• This field allows division operations that might not be possible within \(R\).

By exploring quotient fields and their representations, students grasp how to extend the set of numbers they work with, gaining insights into algebraic structures.

Greatest Common Divisor

The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. In a Unique Factorization Domain (UFD), every non-zero element factors uniquely into prime elements, allowing us to calculate the GCD more straightforwardly.

Here’s why GCD is important:

• It helps in simplifying fractions by removing common factors.
• In UFDs, finding the GCD confirms that a fraction is reduced to its simplest form.

Calculating the GCD in fractions ensures unique and minimal representation, aiding in clear mathematical expressions.

Unit of a Ring

A unit in a ring \(R\) is an element that possesses a multiplicative inverse. In simpler terms, if you multiply a unit by another element, the result is 1 (the multiplicative identity).

Key features of a unit include:

• Units can reverse multiplication effects, essential for solving equations.
• The existence of units helps maintain equality, as seen in unique representations up to units.

Understanding units aids in discerning structure within a ring, recognizing elements that can simplify or transform expressions.

Prime Factorization

Prime factorization breaks down numbers into a product of primes. In a Unique Factorization Domain, each element has a unique prime factorization, except for the order of factors.

Why prime factorization matters:

• It’s foundational for identifying number properties and relationships.
• Helps in simplifying expressions and finding GCDs.

For algebraic structures, prime factorization ensures no overlap of prime factors, allowing clear identification of number components. Comprehending this concept is crucial for tackling many algebraic problems effectively.