5. Let R be a commutative ring with identity. Let f: R[x] -f(p(x)) = E, ai, where p(x) = ao+a1x + a2x² + ... + amx™m. Prove that f isa homomorphism of rings. Notice that f(p(x)) = p(1), where 1 = 1R E R.→ R be defined by4 4. Risc

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Answered: 5. Let R be a commutative ring with…

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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3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[x]/I provides an abstract construction of the ring R[ɛ]. (a) Define a function : R[ɛ] → R[x]/I by (a + b) = a + bx + I Verify that is a ring homomorphism. (b) Consider a polynomial f(x) = a + bx + c2x² + €3x³ + ….. +Cnx² € R[x]. Show that f(x) + I = a + bx + I are equal as equivalence classes. (c) Using part (b), explain why the homomorphism from part (a) is surjective.
4. (a) Let S be a commutative ring with 1 and let s E S. Assume that sn = 0 for some integer n ≥ 2. Show 1 - s is a unit of S. Hint: Expand the product (1s) (1+s+...+ sn−1). (b) Use (a) above to find a commuative ring R with 1 and a non-constant polynomial ƒ € R[x] such that f is a unit of R[x].
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.
If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring operations? How would you prove that? If not, what could be a counter-example?
Let G = Z16X Z4. (i) Construct a surjective homomorphism : G → Z8. (ii) What is ker()? (iii) Show that there is no surjective homomorphism y: G → Z8 × Z8.
10. Consider the irreducible polynomial f = X' + X³ + X² + X + 1 in Z[X]. Let a = [X] in the ring Z[X]/(f). Express (a + 1)(a² + 1) in the form aza + aza² + aa + ao where a; E Z.
Let D: Q[x] → Q[x] akx²+...+a₁x + að ↔ kakxk−1 +...+a₁ be the derivative map on rational polynomials. Is D a ring homomorphism? Is it an isomorphism?
Define a ring homomorphism 4: R[x] → RX R, (ƒ(x)) = (fƒ(0), ƒ(1)) In other words, y maps f(x) to its evaluations at x = and at x = = 1. (a) Describe the kernel of y, in terms of polynomials having certain roots/zeroes. (b) Find polynomials f(x) and g(x) in R[x] such that f(0) = 1, f(1) = 0, and g(0) = 0, g(1) = 1 (Note: There are many different options!) Let a, b € R, and define h(x) = af (x) + bg(x). Show that y(h(x)) = (a,b). (c) Using part (b), explain why is surjective. (d) What does the 1st Isomorphism Theorem say, when applied to ?

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5. Let R be a commutative ring with identity. Let f: R[x] → R be defined by f(p(x)) = Eo ai, where p(x) = ao + a1x+ a2r² +... a homomorphism of rings. Notice that f(p(x)) = p(1), where 1 = 1R E R. + amx. Prove that f is