Exercise 1. Let R and S be rings with identity, p: R → S a homomorphism of rings with p(1r) = 1s, s e S and f e R[x]. Compute p,(f) if (a) R=Z,S = Z, p(n) = [n]5 for all ne Z, s = [2]; and f = r³ + 2x + 1. (b) R= R, S = C, p(r) = r for all r e R, s = i and ƒ = r³ + 2x² + x +2. (c) R = C, S = C[x], p(a+bi) = a – bi for all a, b e R, s =r and f = ix³ – xª + (2 – i)x³ + x² – ix + (3 + 2i).
Exercise 1. Let R and S be rings with identity, p: R → S a homomorphism of rings with p(1r) = 1s, s e S and f e R[x]. Compute p,(f) if (a) R=Z,S = Z, p(n) = [n]5 for all ne Z, s = [2]; and f = r³ + 2x + 1. (b) R= R, S = C, p(r) = r for all r e R, s = i and ƒ = r³ + 2x² + x +2. (c) R = C, S = C[x], p(a+bi) = a – bi for all a, b e R, s =r and f = ix³ – xª + (2 – i)x³ + x² – ix + (3 + 2i).
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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![Exercise 1. Let R and S be rings with identity, p: R → S a homomorphism of rings with p(1r) = 1s, s e S
and f e R[x]. Compute p,(f) if
(a) R= Z,S = Z, p(n) = [n]s for all ne Z, s = [2]5 and f = x³ + 2x + 1.
(b) R= R, S = C, p(r) = r for all r e R, s = i and ƒ =x³ + 2x² + x + 2.
(c) R = C, S = C[x], p(a+ bi) = a – bi for all a,be R, s = x and f = ix³ – xª + (2 – i)x³ + x² – ix + (3+ 2i).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F610b612b-fcf5-4112-b9de-99ecbe09de4b%2F2a83351c-8a85-4bc6-b6a9-c7082c6efb72%2F09iydp6_processed.png&w=3840&q=75)
Transcribed Image Text:Exercise 1. Let R and S be rings with identity, p: R → S a homomorphism of rings with p(1r) = 1s, s e S
and f e R[x]. Compute p,(f) if
(a) R= Z,S = Z, p(n) = [n]s for all ne Z, s = [2]5 and f = x³ + 2x + 1.
(b) R= R, S = C, p(r) = r for all r e R, s = i and ƒ =x³ + 2x² + x + 2.
(c) R = C, S = C[x], p(a+ bi) = a – bi for all a,be R, s = x and f = ix³ – xª + (2 – i)x³ + x² – ix + (3+ 2i).
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