Problem 20 - A First Course in Abstract Algebra

Question

Let \(D\) be a UFD. An element \(c\) in \(D\) is a least common multiple (abbreviated lem) of two elements \(a\) and \(b\) in \(D\) if \(a|c, b| c\) and if \(c\) divides every element of \(D\) that is divisible by both \(a\) and \(b\). Show that every two nonzero elements \(a\) and \(b\) of a Euclidean domain \(D\) have an lem in \(D\). [Htnt: Show that all common multiples, in the obvious sense, of both \(a\) and \(b\) form an ideal of \(D .]\)

Step-by-Step Solution

Answer

In a Euclidean domain, all common multiples form an ideal, generated by the least common multiple.

Step 1: Understand the Problem

We need to prove that for any two non-zero elements \(a\) and \(b\) in a Euclidean domain \(D\), there exists a least common multiple (lem) of these elements. The properties of lem include divisibility by \(a\) and \(b\), and divisibility of any common multiple of \(a\) and \(b\) by the lem.

Step 2: Define the Set of Common Multiples

Consider the set \( S = \{ d \in D \mid a \mid d \text{ and } b \mid d \} \) of all common multiples of \(a\) and \(b\). Our aim is to investigate the properties of this set in the context of an ideal of \(D\).

Step 3: Show That \(S\) Forms an Ideal

For \(S\) to be an ideal, it must be closed under addition and multiplication by any element of \(D\). 1. **Addition**: If \(d_1, d_2 \in S\), then \(a\mid d_1\) and \(a\mid d_2\), so \(a\mid (d_1 + d_2)\). Similarly, \(b\mid (d_1+d_2)\), thus \(d_1 + d_2 \in S\).2. **Multiplication**: For \(d \in S\) and any \(r \in D\), \(a\mid d\) implies \(a\mid rd\). Similarly, \(b\mid d\) implies \(b\mid rd\). Hence \(rd \in S\). Hence, \(S\) is an ideal in \(D\).

Step 4: Employ the Euclidean Domain Property

Since \(S\) is an ideal in the Euclidean domain \(D\), which is also a principal ideal domain (PID), there exists an element \(c\) in \(D\) such that \(S = (c)\). This suggests that every element of \(S\) is divisible by \(c\), and \(c\) is the minimal generator of the ideal.

Step 5: Verify the Least Common Multiple Property

To ensure that \(c\) is an lem of \(a\) and \(b\), it must satisfy:1. \(a \mid c\) and \(b \mid c\), which follows from \(c \in S\) since \(c\) generates \(S\).2. If any element \(m\) in \(D\) is a common multiple of \(a\) and \(b\), i.e., \(m \in S\), then \(c\) divides \(m\) because \(S\) is generated by \(c\). Hence, \(c\) satisfies the properties of the least common multiple.

Key Concepts

Least Common Multiple

The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both. In the context of a Euclidean domain, this concept is quite significant. A Euclidean domain is a type of ring that allows division with a remainder, similar to integers. For two elements, say \(a\) and \(b\), in such a domain, to have a least common multiple \(c\), \(c\) must meet certain conditions.

• First, \(c\) must be divisible by both \(a\) and \(b\). This means that \(a | c\) and \(b | c\), where the notation \(|\) means "divides" without a remainder.
• Secondly, \(c\) must divide any other element that is a common multiple of \(a\) and \(b\). This ensures \(c\) is the 'least' or smallest multiple that \(a\) and \(b\) share.

In a Euclidean domain, the beauty of the LCM is that it can always be found because common multiples form an ideal, which can be generated by a single element, ensuring the existence of the LCM as we prove in the exercise. It shows the LCM isn't just an abstract concept, but a very concrete and essential one with powerful applications.

Unique Factorization Domain

A unique factorization domain (UFD) is a mathematical structure where every element can be broken down uniquely into prime factors, similar to how integers can be factored. In simpler terms, this means:

• Every element (except zero and units) can be written as a product of prime elements.
• This factorization into primes is unique, i.e., even if the order of prime factors may differ, the actual primes are the same.

This property is crucial because it assures us that all elements behave predictably. For instance, in polynomial rings, which are UFDs, each polynomial can be factored into irreducible polynomials in a unique way. This predictable nature is essential for defining other abstract concepts like the least common multiple because it ensures a consistent way of handling divisibility and factorization without ambiguous outcomes.

Principal Ideal Domain

A principal ideal domain (PID) is a special type of ring where every ideal is generated by a single element, called a principal element. This property simplifies many mathematical operations and concepts because:

• Any subset of a PID formed by elements sharing common properties, like divisibility, can be represented by just one generating element of this subset.
• Euclidean domains are always PIDs, and the proof of the LCM in the exercise hinges on this property. For instance, the set of common multiples of two elements forms an ideal that can be described by a single element.

Understanding PIDs is critical because they allow us to scale down the complexity of problems involving divisibility and linear combinations. They provide a streamlined way to handle ideals, making tasks like finding least common multiples more feasible and less cluttered with unnecessary elements.