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 \(R[x]\).

Answered: 6. Let I be an ideal of a ring R and…

Advanced Engineering Mathematics

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ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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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 =