Problem 21 - A Course in Galois Theory

Question

Suppose that \(R\) is an integral domain with the property that every non-empty set \(B\) of non-zero elements has a highest common factor of the form \(\gamma_{1} b_{1}+\cdots+\gamma_{n} b_{n}\), with \(b_{1}, \ldots, b_{n}\) in \(B\) and \(\gamma_{1}, \ldots, \gamma_{n}\) in \(R\). Show that \(R\) is a principal ideal domain.

Step-by-Step Solution

Answer

Since every ideal in \( R \) can be generated by a single element, \( R \) is a principal ideal domain (PID).

Step 1: Understanding the Definitions

First, let's understand the terms we're dealing with. An integral domain is a non-zero commutative ring where the product of any two non-zero elements is non-zero. A principal ideal domain (PID) is an integral domain where every ideal is generated by a single element.

Step 2: Recognizing the Given Property

We are given that for every non-empty set \( B \) of non-zero elements in \( R \), there exists a highest common factor (HCF) of the form \( \gamma_1 b_1 + \cdots + \gamma_n b_n \), which means this HCF can be expressed as a linear combination of elements of \( B \). This is reminiscent of the Euclidean algorithm used for finding gcds in integers, reducing any ideal containing a set to being generated by a single element.

Step 3: Proving Every Ideal is Principal

Now, to show that \( R \) is a PID, we need to demonstrate that every ideal in \( R \) is generated by a single element. Consider any ideal \( I eq (0) \). By the property of HCFs in \( R \), take any non-zero element \( a \in I \). Since the set \( \{ a \} \) in \( R \) has a HCF expressible as a linear combination, the ideal generated by this element must encompass all elements in \( I \).

Step 4: Using the Property to Conclude

By the given property, select any two non-zero elements \( x, y \) in \( I \). Their HCF, expressed as \( \gamma_1 x + \gamma_2 y \), is also a generator of the ideal generated by \( x \) and \( y \). Extend this to all elements of \( I \), so you have an HCF that can be taken as a generator for the entire ideal \( I \). Thus every non-zero ideal of \( R \) is generated by a single element.

Key Concepts

Integral Domain

An integral domain is a fundamental concept in algebra, serving as a building block for understanding more advanced topics like fields and rings. At its core, an integral domain is a type of non-zero ring. However, two crucial properties distinguish it:

• Commutativity: This means that for any elements \(a\) and \(b\) in the domain, the equation \(a \cdot b = b \cdot a\) holds.
• No zero divisors: In an integral domain, if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\). This property ensures that the multiplication behaves as expected, similarly to the integers.

These properties make integral domains a natural extension of our familiar number systems like the integers, minus complications like division by zero. Understanding integral domains helps us explore richer structures in algebra and eventually leads us to comprehend fields and polynomial rings.

Highest Common Factor

The Highest Common Factor (HCF), also known as the greatest common divisor (GCD), is an important concept when working with numbers in mathematics. Unlike typical numbers, when dealing with rings like integral domains, the concept expands.

• Definition in integral domains: The HCF of a set of elements is a common factor that divides each element in the set without leaving a remainder.
• Role of HCF: In integral domains, if the HCF of a set is expressible as \(\gamma_1 b_1 + \dots + \gamma_n b_n\), this indicates that a special linear combination of elements forms the HCF. This reflects a broader ability to decompose and reconstruct element relationships within the domain.

The ability to express the HCF as a linear combination of elements is what allows us to solve problems related to ideal theory, particularly in understanding structure and division within rings.

Linear Combination

A linear combination is a foundational concept in both algebra and linear algebra. It involves expressing one element as a sum of multiples of a list of other elements. In formal terms, given elements \(b_1, b_2, \ldots, b_n\) from a set and coefficients \(\gamma_1, \gamma_2, \ldots, \gamma_n\) from a ring, the expression \(\gamma_1 b_1 + \gamma_2 b_2 + \cdots + \gamma_n b_n\)is called a linear combination of the elements \(b_1, b_2, \ldots, b_n\).

• Importance in algebra: Linear combinations allow us to express complex ideas like span and basis, critical in vector spaces and modules.
• Essential for solving equations: Linear combinations underpin methods for finding solutions to linear equations and determining properties like independence.

Understanding linear combinations is pivotal for computing HCFs in integral domains and exploring algebraic structures, making them a cornerstone in mathematical theory and practical application.

Ideal

In ring theory, an ideal is a fundamental concept, analogous to normal subgroups in group theory. It is a special subset of a ring that absorbs multiplication by elements in the ring. Here's how ideals function:

• Absorption property: For any element \(x\) in the ideal \(I\) and any ring element \(r\), the product \(r \cdot x\) must also be in \(I\).
• Closed under addition and subtraction: If \(x, y \in I\), then both \(x + y\) and \(x - y\) are also in \(I\).

Principal ideals, which are generated by a single element \(a\), are particularly interesting because they simplify the structure of the ring. In a ring \(R\), the set of all multiples of a single element forms a principal ideal, denoted \((a)\). Principal Ideal Domains (PIDs) take this a step further, requiring that every ideal be principal, simplifying their structure and making them more akin to the integers. Understanding ideals is crucial for exploring domains and facilitates the foundational understanding of how algebraic structures can be generated and analyzed.