6. Let I be an ideal of a ring R and let : R→→ R/I be the canonical homomorphism (a) =a+I. Define : R[x] → (R/I) [x] by î (Σa¿x²); Σ(ai + I)x¹: you can assume î is a homomorphism 슈 (a) Prove is onto 1 =
6. Let I be an ideal of a ring R and let : R→→ R/I be the canonical homomorphism (a) =a+I. Define : R[x] → (R/I) [x] by î (Σa¿x²); Σ(ai + I)x¹: you can assume î is a homomorphism 슈 (a) Prove is onto 1 =
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
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![6. Let \( I \) be an ideal of a ring \( R \) and let \( \pi: R \to R/I \) be the canonical homomorphism \( \pi(a) = a + I \). Define \( \hat{\pi}: R[x] \to (R/I)[x] \) by \( \hat{\pi} \left( \sum a_i x^i \right) = \sum (a_i + I) x^i \); you can assume \( \hat{\pi} \) is a homomorphism.
(a) Prove \( \hat{\pi} \) is onto.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F905809b3-f86f-4a48-848e-914423a7f462%2Fd8fdb49e-2a6d-484d-9d5b-760503b1b29b%2Fif7pxlr_processed.jpeg&w=3840&q=75)
Transcribed Image Text:6. Let \( I \) be an ideal of a ring \( R \) and let \( \pi: R \to R/I \) be the canonical homomorphism \( \pi(a) = a + I \). Define \( \hat{\pi}: R[x] \to (R/I)[x] \) by \( \hat{\pi} \left( \sum a_i x^i \right) = \sum (a_i + I) x^i \); you can assume \( \hat{\pi} \) is a homomorphism.
(a) Prove \( \hat{\pi} \) is onto.
![(b) Prove \(\ker \hat{\pi} = [I]\) (the ideal of all polynomials with coefficients in \(I\))
(c) Use \(\hat{\pi}\) and the FHT to prove: if \(P\) is a prime ideal of \(R\), then \(P[x]\) is a prime ideal of \(R[x]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F905809b3-f86f-4a48-848e-914423a7f462%2Fd8fdb49e-2a6d-484d-9d5b-760503b1b29b%2Fhp8rmub_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) Prove \(\ker \hat{\pi} = [I]\) (the ideal of all polynomials with coefficients in \(I\))
(c) Use \(\hat{\pi}\) and the FHT to prove: if \(P\) is a prime ideal of \(R\), then \(P[x]\) is a prime ideal of \(R[x]\).
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