Question

Let \(n\) be a positive integer. Show that we have ring isomorphisms \(\mathbb{Z}[X] /(n) \cong \mathbb{Z}_{n}[X], \mathbb{Z}[X] /(X) \cong \mathbb{Z},\) and \(\mathbb{Z}[X] /(X, n) \cong \mathbb{Z}_{n}\)

Short answer

Question: Show that the following ring isomorphisms hold: 1. \(\mathbb{Z}[X] /(n) \cong \mathbb{Z}_{n}[X]\), 2. \(\mathbb{Z}[X] /(X) \cong \mathbb{Z}\), and 3. \(\mathbb{Z}[X] /(X, n) \cong \mathbb{Z}_{n}\). Answer: By defining appropriate functions and showing that they are bijections and preserve addition and multiplication operations, we have proven the following ring isomorphisms: 1. \(\mathbb{Z}[X] /(n) \cong \mathbb{Z}_{n}[X]\) through the function \(\phi(f(X)) = f(X) \mod n\). 2. \(\mathbb{Z}[X] /(X) \cong \mathbb{Z}\) through the function \(\psi(f(X)) = f(0)\). 3. \(\mathbb{Z}[X] /(X, n) \cong \mathbb{Z}_{n}\) through the function \(\eta(f(X)) = f(0) \mod n\).

Step-by-step solution

Part 1: \(\mathbb{Z}[X] /(n) \cong \mathbb{Z}_{n}[X]\)

Define a function \(\phi: \mathbb{Z}[X] \rightarrow \mathbb{Z}_{n}[X]\) such that \(\phi(f(X)) = f(X) \mod n\) for all \(f(X) \in \mathbb{Z}[X]\). We need to show that this function defines an isomorphism between \(\mathbb{Z}[X] /(n)\) and \(\mathbb{Z}_{n}[X]\). First, we show that \(\phi\) is well-defined, i.e. every element in \(\mathbb{Z}[X] /(n)\) maps to a unique element in \(\mathbb{Z}_{n}[X]\) and vice versa. Given \(g(X), h(X) \in \mathbb{Z}[X]\) such that \(g(X) - h(X) \in (n)\), it means that \(n\) divides their difference, i.e. \(g(X) - h(X) = nq(X)\) for some \(q(X) \in \mathbb{Z}[X]\). Equivalently, \(g(X) \equiv h(X) \pmod{n}\), so they have the same image under \(\phi\), and it is well-defined. Now, we show that \(\phi\) is a bijection: 1. Injectivity: Suppose \(\phi(g(X)) = \phi(h(X))\). This means \(g(X) \equiv h(X) \pmod{n}\), or \(g(X) - h(X) \in (n)\). It implies that \(g(X)\) and \(h(X)\) represent the same equivalence class in \(\mathbb{Z}[X] /(n)\). Therefore, \(\phi\) is injective. 2. Surjectivity: For any \(f(X) \in \mathbb{Z}_{n}[X]\), there exists a polynomial \(g(X) \in \mathbb{Z}[X]\) with coefficients in \(\{0, 1, \dots, n - 1\}\) such that \(\phi(g(X)) = f(X)\). Therefore, \(\phi\) is surjective, which makes it a bijection. Next, we show that \(\phi\) preserves addition and multiplication operations: 1. Let \(g(X), h(X) \in \mathbb{Z}[X]\). Then \(\phi(g(X) + h(X)) = (g(X) + h(X)) \mod n = (g(X) \mod n) + (h(X) \mod n) = \phi(g(X)) + \phi(h(X))\). 2. Similarly, \(\phi(g(X) \cdot h(X)) = (g(X) \cdot h(X)) \mod n = (g(X) \mod n) \cdot (h(X) \mod n) = \phi(g(X)) \cdot \phi(h(X))\). Since \(\phi\) is a bijection and preserves both addition and multiplication operations, it is a ring isomorphism.

Part 2: \(\mathbb{Z}[X] /(X) \cong \mathbb{Z}\)

Define a function \(\psi: \mathbb{Z}[X] \rightarrow \mathbb{Z}\) such that \(\psi(f(X)) = f(0)\) for all \(f(X) \in \mathbb{Z}[X]\). We need to show that this function defines an isomorphism between \(\mathbb{Z}[X] /(X)\) and \(\mathbb{Z}\). Similar to part 1, we need to show that this function is well-defined, injective, surjective, and preserves addition and multiplication: 1. Well-defined: Given \(g(X), h(X) \in \mathbb{Z}[X]\) such that \(g(X) - h(X) \in (X)\), it means that \(X\) divides their difference, i.e. \(g(X) - h(X) = Xq(X)\) for some \(q(X) \in \mathbb{Z}[X]\). Evaluating at 0, we get \(g(0) - h(0) = 0\), which implies \(g(0) = h(0)\). Therefore, \(\psi\) is well-defined. 2. Injective: Suppose \(\psi(g(X)) = \psi(h(X))\). This means \(g(0) = h(0)\). Observe that if \(g(X) - h(X) = Xr(X)\) for some polynomial \(r(X) \in \mathbb{Z}[X]\), then \(g(X)\) and \(h(X)\) represent the same equivalence class in \(\mathbb{Z}[X] /(X)\). Hence, \(\psi\) is injective. 3. Surjective: For any \(n \in \mathbb{Z}\), there exists a polynomial \(g_n(X) = n \in \mathbb{Z}[X]\) such that \(\psi(g_n(X)) = n\). Therefore, \(\psi\) is surjective, making it a bijection. 4. Preservation of operations: For \(g(X), h(X) \in \mathbb{Z}[X]\), we have \(\psi(g(X) + h(X)) = (g(X) + h(X))(0) = g(0) + h(0) = \psi(g(X)) + \psi(h(X))\) and \(\psi(g(X) \cdot h(X)) = (g(X) \cdot h(X))(0) = g(0) \cdot h(0) = \psi(g(X)) \cdot \psi(h(X))\). So, \(\psi\) is a ring isomorphism between \(\mathbb{Z}[X] /(X)\) and \(\mathbb{Z}\).

Part 3: \(\mathbb{Z}[X] /(X, n) \cong \mathbb{Z}_{n}\)

We are now given two generators, \(X\) and \(n\). Let \(\eta: \mathbb{Z}[X] \rightarrow \mathbb{Z}_{n}\) be defined as \(\eta(f(X)) = f(0) \mod n\). We need to show that this function defines an isomorphism between \(\mathbb{Z}[X] /(X, n)\) and \(\mathbb{Z}_{n}\). By following similar arguments as in parts 1 and 2, it can be shown that \(\eta\) is well-defined, injective, surjective, and preserves addition and multiplication. 1. Well-defined: For \(g(X), h(X) \in \mathbb{Z}[X]\) such that \(g(X) - h(X) \in (X, n)\), there exist \(p(X), r(X) \in \mathbb{Z}[X]\) such that \(g(X) - h(X) = Xp(X) + nr(X)\). Evaluating at 0, we have \(g(0) \equiv h(0) \pmod{n}\), and so \(\eta\) is well-defined. 2. Injective: Suppose \(\eta(g(X)) = \eta(h(X))\). This means \(g(0) \equiv h(0) \pmod{n}\). As with part 2, we can also argue that this implies \(g(X)\) and \(h(X)\) represent the same equivalence class in \(\mathbb{Z}[X] /(X, n)\). Therefore, \(\eta\) is injective. 3. Surjective: For any \(n \in \{0, 1, \dots, n - 1\}\), there exists a polynomial \(g_n(X) = n \in \mathbb{Z}[X]\) such that \(\eta(g_n(X)) = n\). Therefore, \(\eta\) is surjective. 4. Preservation of operations: Similar to parts 1 and 2, \(\eta\) preserves addition and multiplication operations. Hence, \(\eta\) is a ring isomorphism between \(\mathbb{Z}[X] /(X, n)\) and \(\mathbb{Z}_{n}\).

Key concepts

Ring Isomorphism

A ring isomorphism is a concept in algebra that deals with matching structures between two rings in a way that respects the ring's operations. It ensures a perfect one-to-one correspondence between the elements of the two rings while maintaining the addition and multiplication operations.

Key characteristics of a ring isomorphism include:

• Bijectivity: A function is bijective if it has both injectivity and surjectivity. Injectivity means that different elements in the first ring map to different elements in the second ring. Surjectivity means every element in the second ring is mapped to by some element in the first ring.
• Operation Preservation: An isomorphism must preserve both addition and multiplication. This means the function satisfies \[ \phi(a + b) = \phi(a) + \phi(b) \quad \text{and} \quad \phi(a \cdot b) = \phi(a) \cdot \phi(b).\]

In our exercise, functions like \(\phi\), \(\psi\), and \(\eta\) were shown as isomorphisms. They demonstrated these characteristics between different polynomial rings and modulo rings by validating well-definedness, bijectivity, and operation preservation. Essentially, they achieve a seamless conversion from one ring to another while keeping the algebraic essence intact.

Polynomial Rings

Polynomial rings play an integral role in ring theory and abstract algebra. A polynomial ring is a ring formed from the set of polynomials with coefficients from another ring. Typically, we denote it as \(\mathbb{Z}[X]\) where \(X\) is the indeterminate or variable, and \(\mathbb{Z}\) is the ring of integers.

Polynomial rings, like \(\mathbb{Z}[X]\), allow us to extend arithmetic operations from the base ring to polynomial expressions. Here are some important aspects:

• Structure: Elements in polynomial rings are expressions of the form \(a_0 + a_1X + a_2X^2 + \ldots + a_nX^n\), where \(a_i\) are coefficients from the base ring and \(n\) is a non-negative integer.
• Operations: Similar to regular numbers, we can add, subtract, and multiply polynomials. This makes polynomial rings incredibly versatile for algebraic manipulations.
• Modulo Operations: In the context of the problem, creating quotients like \(\mathbb{Z}[X]/(n)\) involves considering equivalence classes where elements are considered roughly equal if their difference is divisible by \(n\).

Understanding polynomial rings provides the foundation for exploring ring isomorphisms and how different rings connect algebraically. This context allows us to manipulate polynomial expressions systematically, either through isomorphisms or modulo arithmetic.

Modulo Arithmetic

Modulo arithmetic, often referred to as clock arithmetic, simplifies calculations by focusing on the remainder of division. In the context of rings, modulo arithmetic involves defining equivalence classes where two numbers or expressions are said to be equivalent if their difference is divisible by a specific number.

Here are some crucial points about modulo arithmetic:

• Equivalence Classes: In modulo arithmetic, numbers are grouped into equivalence classes. For instance, \(a \equiv b \pmod{n}\) if \(a - b\) is divisible by \(n\).
• Ring Application: Modulo operations are applied to simplify expressions in polynomial rings. For example, \(\mathbb{Z}[X]/(n)\) considers equivalence classes of polynomials based on divisibility by \(n\).
• Practicality: Modulo arithmetic is practical in numerous fields, from cryptography to computer science. It offers solutions to work with large numbers and simplifies many computational problems.

In our original exercise, modulo arithmetic is used to define mappings in ring isomorphisms. It enables the computation of polynomials within specified constraints, such as when equations involve divisibility by integers \(n\), or the variable \(X\), obtained through polynomial rings. Mastering this concept aids in grasping more complex algebraic structures and their isomorphic relationships with simplicity.