More generally, let f(x) be an arbitrary polynomial, and let (f(x)) denote the set of all polynomials that are a multiple of f(x), i.e. the set {f(r)g(x)| g(x) E R[r]}. Show that (f(x)) is an ideal of R[r].
More generally, let f(x) be an arbitrary polynomial, and let (f(x)) denote the set of all polynomials that are a multiple of f(x), i.e. the set {f(r)g(x)| g(x) E R[r]}. Show that (f(x)) is an ideal of R[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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abstract algebra
![**Example 2.68**
In the ring \(\mathbb{R}[x]\), let \(\langle x \rangle\) denote the set of all polynomials that are a multiple of \(x\), i.e., the set \(\{ xg(x) \mid g(x) \in \mathbb{R}[x] \}\).
**Exercise 2.68.2**
More generally, let \(f(x)\) be an arbitrary polynomial, and let \(\langle f(x) \rangle\) denote the set of all polynomials that are a multiple of \(f(x)\), i.e., the set \(\{ f(x)g(x) \mid g(x) \in \mathbb{R}[x] \}\). Show that \(\langle f(x) \rangle\) is an ideal of \(\mathbb{R}[x]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2F32d56e5d-6f7c-49f8-a3e8-23ec2dafc65d%2Fdpzcu9n_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Example 2.68**
In the ring \(\mathbb{R}[x]\), let \(\langle x \rangle\) denote the set of all polynomials that are a multiple of \(x\), i.e., the set \(\{ xg(x) \mid g(x) \in \mathbb{R}[x] \}\).
**Exercise 2.68.2**
More generally, let \(f(x)\) be an arbitrary polynomial, and let \(\langle f(x) \rangle\) denote the set of all polynomials that are a multiple of \(f(x)\), i.e., the set \(\{ f(x)g(x) \mid g(x) \in \mathbb{R}[x] \}\). Show that \(\langle f(x) \rangle\) is an ideal of \(\mathbb{R}[x]\).
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