*Ring Homomarphism 11. Let R and R' be two rings. A mapping f: R→R' iscalled an antihomomorphism, ifƒ (x + y) =f(x) +ƒ(y) and f(xy) =ƒ (y) ƒ (x) \ x, y ɛ R.1Let f, g be two antihomomorphisms of a ring R into R. Prove thatfg: R R is a homomorphism.—

Let x, y ∈ R. Then (g)(x + y) = f(g(x + y)) = f(g(x) + g(y)) = f(g(x)) + f(g(y)) =(fg)(x)+(fg)(y),…

Answered: 11. Let R and R' be two rings. A…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter6: More On Rings

Section6.2: Ring Homomorphisms

Problem 6E

See similar textbooks

Similar questions

Prove that if a subring R of an integral domain D contains the unity element of D, then R is an integral domain. [Type here][Type here]
11. Show that defined by is not a homomorphism.
Construct bijective maps(a) from [0, 1) to (0, 2].(b) from R − {3} to R − {1}(c) from Z to Z − {1, 2}.
8. Let R denote the ring Z[i]/(1 + 3i). (b) Sho (d) Show that R=Z/10z.
6. Let R be a ring and consider RxZ= {(r,n)|r e R, n e Z}. (r,n) + (s,m) = (rts, n+m) (r,n)(s,m) Define = (rs+mr+ns, nm) Show that R x Zis a ring.
B E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) → (Q(√3), +,-), a +b√2+a+b√3. (ii): f: C Rx R, a +bi (a, b). (iii): f: (QxQ, +,-) → (Q(√2), +,-), (x,y) →x+yv2. (iv): f: CR, a +bi+ a² + b². ...
2. be f Let ƒ : R\{3} → R\{5} be defined by f(x) = 5x/(x-3). Show that fis bijective and find f-¹.
2. Prove that f: (R × R, +) → (R,+) given by f(x, y) = x + y is a homomorphism. Is g: (R × R, +) → (R, +) given by g(x, y) = x.y a homomorphism? Explain your answer. (Hint: Check whether f((a, b) + (c,d)) = f(a,b) + f(c,d).)
1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2
Let R be any ring. If f(X) = ao + α₁X + a₂X² + . = ao + a₁X + a₂X² + ... + a₂X¹ € R[X], define the anX" (formal) derivative of f by f'(X) = a₁ + 2a₂X+nan X-1. Prove that for any polynomials f(X), g(X) € R[X], (ƒ(X) + g(X))' = f'(X) + g'(X) (f(X)g(X))' = f(X)g'(X) + f'(X)g(X)
3. Construct a bijection f: [0,1] → (0, 1) UN.
Show that ℝ2 = span([1 1], [1 -1])

SEE MORE QUESTIONS

Recommended textbooks for you

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

Elements Of Modern Algebra

Algebra

ISBN:

9781285463230

Author:

Gilbert, Linda, Jimmie

Publisher:

Cengage Learning,

SEE MORE TEXTBOOKS

11. Let R and R' be two rings. A mapping f: R→R' is called an antihomomorphism, if f(x+y)=f(x) + f(y) and f(xy) =f(y)f(x) \ x, y = R. Let f, g be two antihomomorphisms of a ring R into R. Prove that fg: RR is a homomorphism.