Determine if each of the following maps is a ring homomorphism.(a) f : R → R defined by f(x)%3D-x1 -2= (6 1) a ((b) g : M2(IR) → M2(R) defined1

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Answered: Determine if each of the following maps…

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.
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.
(5) Recall that an element of a ring is idempotent if a² = a. (a) Show that any integral domain can have only 1 and 0 as idempotent elements. (b) Is the converse to the previous statement true? Hint: Consider the ring of continuous functions defined on the interval [0, 1], which we call C([0, 1]). What are the idempotent elements? Is this ring an integral domain?
Let R be a commutative ring with unity and c E R. The map R[X] → R[X] f(X) → R= f(X + c) is an isomorphism of rings.
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 ?
(19) Let om : Z → Zm be the ring homomorphism defined by om (a) remainder of division of a by m. (a) Show that m : Z[x] → Zm[x] defined by om (anx + + a₁x + ao) = om(an)x² + is a ring homomorphism onto Zm[x]. + om (a₁)x+om (ao) = (b) Show that if f(x) = Z[x] and ¯m (f(x)) = Zm[x] both have degree n and σm(f(x)) does not factor in Zm[x] into two polynomials of degree less than n, then f(x) is irreducible Q[x]. (c) Use the previous part to show that x³ + 17x + 36 is irreducible in Q[x].
Abstract algebra 2. Graduate level. Note: let R be any ring and M and N R − modules

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Determine if each of the following maps is a ring homomorphism. (a) f : R → R defined by f(x) %3D -x 1 -2 = (6 1) a ( (b) g : M2(IR) → M2(R) defined 1