2. Consider the following functions. Are these ring homomorphisms? If yes, prove it. If no, provide a counterexample. a) f: ZZ given by f(x) = 3x. b) g: R R given by g(x) = x² - c) h: Z→ M(Z) given by h(a) = [ a 8

Q: (d) Show that g: B → A, g(x) = Vln x is the inverse of f. Why is f-"(x) # -VIn x?

A:

Q: 4. Consider the set of all continuous functions f: R → R: 8(R) = {ƒ :R→R | ƒ is a continuous…

A: Let  f be a  unit in ζ(R). Then by definition, there is a continuous function  g ∈ ζ(R)  such that…

Q: 2. Let D be an integral domain and let D[X] denote the ring of polynomials in the indeterminate X…

A: “Since you have posted a question with multiple sub parts, we will provide the solution only to the…

Q: Let A be an associative algebra. (a) Show that for any a E A the map da : A A defined by da(b) = ab…

A: Let A be an associated algebra.  To prove that for any a∈A, the map da:A→A defined by dab=ab-ba is a…

Q: 4. Let y: R→ S be a ring homomorphism. Prove that ': R[X] → S[X] given by soʻ (ao + a1X + ..a,X") =…

A:

Q: 4. The following polynomial is irreducible in Q[x]: m(x) = x² - 2x²-1 € Q[x] Let's consider the…

A:

Q: Which of the following maps C[X] → C[X] are ring homomorphisms? Select all that apply: O The map…

A:

Q: 2. Let s-{(소)아 2a S = 26 2a Assume that S is a subring of M2(Q). Prove that S is isomorphic to Q[V2]…

A:

Q: Problem 1. Let R be a ring. Construct an isomorphism between¹ R[X,Y]/(XY) and the subring A CR[X] ×…

A:

Q: Problem 7 Let He Vam I tudom S = {(2,x) | x E I} CR², dT (81 Ing/ T= {(0,x) | x 1} CR², o ir aut

A:

Q: a) show that the given function is 1-1 or give a counterexample. b) show that the given function is…

A:

Q: We denote the set of polynomials with real coefficients as R[X]:= {ao + a₁X + a₂X² +.... + a₂X": n€…

A: (.)  Given set of polynomials,and linear map  defined as,    We will show that  is a linear operator…

Q: Evaluate the limit 9 1 - cos(t) 7. | lim t→0 t – 8 i + In(t + 8)j + -k t Enter your answer in i, j,…

A:

Q: Qs: (A) Let R be a ring with identity. Define g:Z Z by g(x) = x. 1, Vx € Z. Is a a homorphism?…

A: Given: Ring R with identity and map g:Z→Z, g(x)=x·1 ∀x∈Z To find: a) Check whether g is homomorphism…

Q: [H] Let 0 be the set of odd integers and consider the function T: Z → 0 defined by r(n) = 2n – 3.…

A: Firstly, we define onto function (Surjective function). Let X and Y are two sets. Then f:X→Y is said…

Q: 3. Consider the ring R[x], and its ideal I = (x²). Then the quotient ring R[T]/I provides an…

A: Using properties of Ring homomorphism we solve this problem.

Q: 14. Find all units of the ring C(R) = {ƒ : R → Rf is continuous}, where the operations are the usual…

A: A unit is an element which has multiplicative inverse. Using this definition we find the units in…