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

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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14. Find all units of the ring C(R) = {ƒ : R → Rf is continuous}, where the operations are the usual point-wise addition and multiplication of functions. Also, give an example, if there exists, of a zero-divisor.
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3. Determine whether the following mappings are ring homomorphisms: f: Q- Q defined by f(x)= |x| for all x e Q. a) b) c) g: CM₂ (R) defined by g (a + bi) = a b -b a h: Z[√2] → Z[√2] defined by h( a + b √2) =) a - b √2
Each of the quotient rings R/I on the left is isomorphic to one of the rings S on the right. Match each ring with its isomorphic partner, and prove that they really are isomorphic by constructing a surjective ring homomorphism p: RS with kernel I. R/I S A) Z[X]/(8, 12, X) 1. ZZ 2. Q 3. R B) Q[X]/(X² - 2) 4. C 5. F4 C) R[X]/(X-√2) 6. F8 7. ZA 8. Zg D) R[X]/(x²+X+2) 9. {a+b√2 a,bQ} 10. a {( 8 b ) | ab € F₁ } a E) R[X]/(X²) F) Z₁₂[X]/(X + X²+1) a 11. | a, b € F8 0 a 12. {( o b) | ab € R } a 0 a In examples 5, 6, 10, and 11 the symbols F4 and Fg denote a field of 4 and a field of 8 elements respectively. In examples 10, 11, and 12 the ring operations are addition and multiplication of matrices. Some methods you might use to try and match up R/I and S: (i) guess, (ii) think about representatives in the quotient ring, and what the multiplication rules are and try and match that up with something in the S column, (ii) try and make up a homomorphism from R to somewhere that would have elements…
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Number 3
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Let R be a commutative ring with unity and let S be a nonempty subset of R. Show that I(S) := {f(x) = R[x] : f(a) = 0, for all a € S} is an ideal of R[x].
Let R he a ring and f: RR be defined by f(z) = 1". %3D Check Al that are correct. O tis a group homomorphism for (R= Z4, +). %3D fis a ring homomorphism for (R = Z4, t.). fis not onto when R = Z7. |3D O tis one-to-one when R= Zg. %3D tis not onto when R = Zz. %3D tis a group homomorphism for (R = Z2, +).
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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 =