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

If a0 in a field F, prove that for every bF the equation ax=b has a unique solution x in F. [Type here][Type here]

1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

Let R be the set of all continuous functions. It is known that (R, +,×) is a ring where + and × are both performed pointwise. Answer the following questions.a. Let f(x) = x2 and g(x) = 3x. What is f + g?b. Let f(x) = x2 and g(x) = 3x. What is f × g?c. Prove that f given by f(x) = 1 is 1R

9. Iff js a homomorphism from a ring R into a ring S then show that f(0)=0 1 :: f(- a) = -f(a), V a e R.

7. Let R be a commutative ring and let F = F(R, R) be the set of functions f : R R. Functions in F can be added and multiplied pointwise using the operations from R, i.e. define (S + 9)(x) = f(x) + g(x) (f9)(2) = f(a)g(x) (a) Prove F is a commutative ring. (b) Let I denote the identity function 1(r) = r in F. Prove that the map p : R[X] → F given by p(a, X" +. + a1 X + ao) = a," + . + a1l + ao ... is a ring homomorphism, where R[X] is the ring of polynomials with coefficients in R. 1 of 2 MATH 4107: Homework 10 (c) Give an example showing that p is in general not injective (hint: try R= Z/2Z). Find ker(p) when R = Z/pZ where p is prime. Note that the fact that p is not injective means that one cannot in general view polynomials in R[X] as R-valued functions on R, which is in sharp contrast to the case of Z[X], R[X], or C[X].

Explain why the polynomial rings R[r] and C[x] are not isomorphic.

4. Consider the set of all continuous functions f: R → R: 8(R) = {ƒ :R→R | ƒ is a continuous function}, This set forms a ring under the usual addition/subtraction/multiplication of functions. The additive identity element in this ring is the constant function 0, and multiplicative identity element is the constant function 1. (a) Let f: R → R be a continuous function. Show that ƒ is a unit in C(R) if and only if the fucntion f has no zeroes. (b) Show that the ring C(R) contains zero divisors. That is, give an example of two non-zero continuous functions f,g: R→ R such that their product fg is the zero function.