Problem 17 - A First Course in Abstract Algebra

Question

Prove or disprove the following statement: If \(\nu\) is a Euclidean norm on Euclidean domain \(D\), then \(\mid a \in\) \(D \mid v(a)>v(1)] \cup\\{0\\}\) is an ideal of \(D .\)

Step-by-Step Solution

Answer

The statement is false; the set is not an ideal.

Step 1: Understand Euclidean Norm and Domain

A Euclidean domain is a type of ring where division with remainder is possible. The Euclidean norm \(u\) assigns a non-negative integer to each non-zero element of \(D\), such that for any two elements \(a\) and \(b\), there exist \(q\) and \(r\) in \(D\) with \(a = bq + r\) and either \(r = 0\) or \(u(r) < u(b)\).

Step 2: Define the Set

Consider the set \( I = \{ a \in D \mid u(a) > u(1) \} \cup \{0\} \). This set consists of all elements \(a\) in \(D\) such that their Euclidean norm is greater than \(u(1)\), along with the zero element.

Step 3: Check if I is an Ideal - Closure under Addition

An ideal must be closed under addition. Take any \(x, y \in I\). If \(x = 0\), then \(x + y = y\), which is in \(I\). If both \(x, y eq 0\) and \(u(x) > u(1)\), \(u(y) > u(1)\), it is not guaranteed that \(u(x+y) > u(1)\). For example, if \(x = y\), then \(x + (-y) = 0 ot\in I\). Thus, \(I\) is not closed under addition.

Step 4: Check if I is an Ideal - Closure under Multiplication by D

An ideal must be closed under multiplication by any element in \(D\). Take any \(a \in I\) and \(d \in D\). If \(a = 0\), then \(da = 0 \in I\). If \(u(a) > u(1)\), we need that \(u(da) > u(1)\). However, for \(d = 1\), \(da = a\) which satisfies this, but for any \(d\) such that \(u(d) < 1\), \(u(da)\) could be less than \(u(1)\), violating this property.

Step 5: Conclusion

To be an ideal, a set must satisfy both closure under addition and under multiplication by elements of \(D\). Since the set \(I\) fails closure under addition, \(I\) is not an ideal of \(D\). Thus, the statement given is false.

Key Concepts

Euclidean norm

The Euclidean norm is an essential concept in understanding Euclidean domains. But what is it exactly? Imagine it as a way to measure the 'size' of elements within a domain. If you have a non-zero element, the Euclidean norm assigns a positive integer to it. The norm is denoted typically as \( u(a) \) for an element \( a \).

Why is it important? It helps in the division process similar to how we might divide numbers. Given two elements \( a \) and \( b \) in the domain, there are integers \( q \) (quotient) and \( r \) (remainder) such that \( a = bq + r \). Now, either \( r \) is zero, or its norm \( u(r) \) is smaller than that of \( b \).

This mathematically facilitates the process of division and is what makes such domains special - allowing operations similar to integer division, even in abstract algebra settings.

ideal

An ideal provides structure within a ring. It's like a special subset of the ring, respecting the rules of the ring through specific properties. But what exactly makes a subset an ideal?

Imagine you have a set of elements. For it to qualify as an ideal, two things must happen:

• Closure under addition: If you take any two elements from this set and add them together, the result should still belong to the set.
• Closure under multiplication by the ring: If you multiply any element of the subset by any element in the ring, the result should still be in the set.

These conditions ensure that the operations of the ring don't force elements out of the subset. Essentially, executing these operations within an ideal results in outcomes that remain contained within it.

ring

A 'ring' in mathematical terms is a set equipped with two binary operations, usually called addition and multiplication. These operations mimic arithmetical ones, such as addition and multiplication of integers. Let's break this down into simpler terms.

In a ring, you can:

• Add elements: The addition is both commutative (\( a + b = b + a \)) and associative (\( a + (b + c) = (a + b) + c \)). There also exists an additive identity (\( 0 \)), meaning any element added to zero is unchanged.
• Multiply elements: This operation is associative but does not require commutativity. This means \( a \times b \) is not necessarily \( b \times a \).

In simpler terms, a ring is a structured, abstract mathematical concept that allows consistent algebraic operations, much like how integers operate with our conventional arithmetic.

division with remainder

The concept of division with remainder is foundational to understanding the structure of Euclidean domains. It's like how we divide numbers in everyday calculations.

Here's how it works in a Euclidean domain: For any two elements \( a \) and \( b \) within the domain (with \( b eq 0 \)), the element \( a \) can be expressed as \( a = bq + r \). Here:

• \( q \) is called the quotient.
• \( r \) is the remainder.

The unique aspect is:

• \( r \) is either zero or has a Euclidean norm smaller than \( b \).

This process is akin to how you might do long division with integers, ensuring that no matter what is being divided, what's "leftover" (the remainder \( r \)) is smaller in terms evoked by the Euclidean norm than what you started dividing by (\( b \)). Playing within these rules keeps the process consistent in abstract settings.