2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime ideal of R[x].
2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime ideal of R[x].
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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Please include a rationale for each step.
![**Problem 2:**
Let \( R \) be a commutative ring with unity. If \( I \) is a prime ideal of \( R \), prove that \( I[x] \) is a prime ideal of \( R[x] \).
**Explanation:**
In this problem, you're asked to demonstrate that if you start with a prime ideal \( I \) in a commutative ring \( R \), the set \( I[x] \), formed in the polynomial ring \( R[x] \), is also a prime ideal. This involves showing that for any polynomials \( f(x), g(x) \in R[x] \), if their product is in \( I[x] \), then at least one of them must be in \( I[x] \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F174d7d89-86f6-4fd3-817b-704de3f40cca%2F9e1cc836-9755-458c-9784-b89c27e3c887%2F3b6uvoh_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 2:**
Let \( R \) be a commutative ring with unity. If \( I \) is a prime ideal of \( R \), prove that \( I[x] \) is a prime ideal of \( R[x] \).
**Explanation:**
In this problem, you're asked to demonstrate that if you start with a prime ideal \( I \) in a commutative ring \( R \), the set \( I[x] \), formed in the polynomial ring \( R[x] \), is also a prime ideal. This involves showing that for any polynomials \( f(x), g(x) \in R[x] \), if their product is in \( I[x] \), then at least one of them must be in \( I[x] \).
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