1. Consider the ring Z[r]. Prove that the ideal (2, x) = {2f(x)+rg(x) : f(x), g(x) E Z[r]} is not a principal ideal; that is, show that (2, r) (p(x)) for any p(x) E Z[r].
1. Consider the ring Z[r]. Prove that the ideal (2, x) = {2f(x)+rg(x) : f(x), g(x) E Z[r]} is not a principal ideal; that is, show that (2, r) (p(x)) for any p(x) E Z[r].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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absract algebra
![**Problem 1**
Consider the ring \( \mathbb{Z}[x] \). Prove that the ideal \(\langle 2, x \rangle = \{2f(x) + xg(x) : f(x), g(x) \in \mathbb{Z}[x]\}\) is not a principal ideal; that is, show that \(\langle 2, x \rangle \neq \langle p(x) \rangle\) for any \(p(x) \in \mathbb{Z}[x]\).
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**Explanation:**
The problem is asking to demonstrate that the ideal generated by 2 and \(x\) in the polynomial ring with integer coefficients, \(\mathbb{Z}[x]\), cannot be created by a single polynomial \(p(x)\).
The set notation describes the elements of the ideal: any element is a combination of \(2\) multiplied by some polynomial \(f(x)\) and \(x\) multiplied by another polynomial \(g(x)\), where both \(f(x)\) and \(g(x)\) are from \(\mathbb{Z}[x]\).
The task is to prove that no single polynomial from \(\mathbb{Z}[x]\) can generate all elements of the ideal \(\langle 2, x \rangle\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2F42dbd7c9-0c70-45c8-8877-c025ef033b16%2F7yxin01_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 1**
Consider the ring \( \mathbb{Z}[x] \). Prove that the ideal \(\langle 2, x \rangle = \{2f(x) + xg(x) : f(x), g(x) \in \mathbb{Z}[x]\}\) is not a principal ideal; that is, show that \(\langle 2, x \rangle \neq \langle p(x) \rangle\) for any \(p(x) \in \mathbb{Z}[x]\).
---
**Explanation:**
The problem is asking to demonstrate that the ideal generated by 2 and \(x\) in the polynomial ring with integer coefficients, \(\mathbb{Z}[x]\), cannot be created by a single polynomial \(p(x)\).
The set notation describes the elements of the ideal: any element is a combination of \(2\) multiplied by some polynomial \(f(x)\) and \(x\) multiplied by another polynomial \(g(x)\), where both \(f(x)\) and \(g(x)\) are from \(\mathbb{Z}[x]\).
The task is to prove that no single polynomial from \(\mathbb{Z}[x]\) can generate all elements of the ideal \(\langle 2, x \rangle\).
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