Problem 12 - Algebra: Pure and Applied

Question

Let \(I\) be a nontrivial ideal in \(\mathbb{Z}[i]\). Show that \(\mathbb{Z}[i] / I\) is a finite ring.

Step-by-Step Solution

Answer

123][𝑖]/𝑰• is finite because it consists of cosets formed from a finite number of elements due to the nt• norm.

Step 1: Understanding Ideals in  [𝑍][𝑖]

In the ring of Gaussian integers ••𝑍][𝑖], an ideal 𝑰• can consist of all integer combinations of a specific Gaussian integer 𝑎+𝑏𝑖• . A nontrivial ideal 𝑰• means 𝑰• is not the zero ideal, and have• includes at least one non-zero element.

Step 2: Analyze Cosets of the Ideal

The ring 𝑍][𝑖]/𝑰• is formed by the cosets of 𝑰• . Each coset is of the form 𝑎+𝑏𝑖+𝑰• where 𝑎+𝑏𝑖• is a generator of𝑰• . Verify that distinct integers or cosets satisfy 𝑎+𝑏𝑖• ∉ 𝑰• . Thus, the number of distinct cosets corresponds to positive integers [𝑍][𝑖]/𝑛•.

Step 3: Determine Finiteness of Cosets

Each coset /𝑖][𝑖]• can be represented by varying 𝑎+𝑏𝑖• from 𝑰• . Thanks to the finite nature of multiples, these distinct combinations group comfortably in a finite tally of cosets.

Step 4: Relate with Norm Function

Consider the norm, 𝑁(𝑎+𝑏𝑖)=𝑎²+𝑏²• . If the norm of 𝑎+𝑏𝑖• is 𝑛• , any element in 𝑰• must have its norm as a multiple of 𝑛• . Since the norm function is a mapping of 𝑎+𝑏𝑖𝑜𝑡𝑟• integers, it implies finitely many coset members with reduced form.

Step 5: Conclude on Ring Finiteness

The number of coset representatives  ]/𝑰• is finite. Therefore, 𝑍][𝑖]/𝑰• is a finite ring.

Key Concepts

Ideals

An ideal in the context of rings, including the ring of Gaussian integers \(\mathbb{Z}[i]\), is an important element. It consists of all integer combinations of a specific element from the ring. When we refer to a nontrivial ideal, we exclude the zero ideal, which contains only the number zero. Instead, we work with ideals that include at least one non-zero element. Ideals in \(\mathbb{Z}[i]\) maintain the essential property that, if you multiply any element from \(\mathbb{Z}[i]\) by an element of the ideal, the result will also belong to the ideal itself. This creates a stable structure where members are predictable by their defining properties.

Cosets

Cosets are fundamental in understanding quotient rings like \(\mathbb{Z}[i] / I\), where \(I\) is an ideal. In essence, cosets partition a ring into non-overlapping subsets. Each coset can be expressed in the form \(a + bi + I\), where \(a + bi\) is a representation of any member of \(\mathbb{Z}[i]\), and \(I\) is the ideal that helps form these coset families. Importantly, two elements belong to the same coset if their difference is in the ideal \(I\). For any ring directly related to cosets, if the number of cosets is countable, it implies certain finiteness in the properties of the overall ring structure. This finiteness assures us that the set of distinct cosets—and hence the ring \(\mathbb{Z}[i] / I\)—is limited.

Gaussian Integers

Gaussian integers are numbers of the form \(a + bi\), where \(a\) and \(b\) are integers, and \(i\) represents the square root of \(-1\). They form a unique mathematical ring, denoted \(\mathbb{Z}[i]\). This ring behaves similarly to the integers \(\mathbb{Z}\) but extends into the complex plane. Properties of Gaussian integers include closure under addition, subtraction, and multiplication. They are foundational in establishing ideals and examining their behavior in the ring \(\mathbb{Z}[i]\). Gaussian integers help depict structures visually through lattice points in the complex plane, where each point corresponds to a Gaussian integer.

Norm Function

The norm function for Gaussian integers is defined as \(N(a + bi) = a^2 + b^2\). This mathematical operation maps a Gaussian integer to a non-negative integer. The norm is essential because it provides insights into the size and divisibility properties of Gaussian integers within the ring. When an ideal \(I\) in \(\mathbb{Z}[i]\) is described by a Gaussian integer with a particular norm \(n\), it implies that elements in \(I\) have norms that are multiples of \(n\). Besides confirming structural properties like finiteness, the norm function allows us to assess the uniqueness of cosets and their representatives. As a direct result, the norms defining these ideals suggest that there are finitely many reduced forms, further confirming the finite nature of \(\mathbb{Z}[i] / I\).