Let
Prove that
Is
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Elements Of Modern Algebra
- 12. Consider the mapping defined by . Decide whether is a homomorphism, and justify your decision.arrow_forward14. Let be a ring with unity . Verify that the mapping defined by is a homomorphism.arrow_forward11. 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: R R is a homomorphism.arrow_forward
- 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 8arrow_forwardUse First Isomorphism Theorem to prove that R/ZE S'.arrow_forwardProve that the dual of l' is isometric to 10⁰.arrow_forward
- 3. Let mappings F= (F1, F2) R² → R² and G = (G1, G2): R² → R² be defined by F₁(x1, x2) = x² + x2, G₁(y1, y2)=sin(y2), F2(x1, x2, x3) = x1 + x2, G2(y1, y2) = cos(y1). (i) Find the composition mapping GoF: R2 → R². (ii) By using the chain rule, find the derivative of the mapping Go F, that is [5 Marks] D(GF)(x1, x2). [15 Marks] (iii) Give reasons why the mapping GoF is differentiable at every x = R². [5 Marks]arrow_forwardLet I = {[xx, y = R} and J = {[²] 1z € R} E Consider the ring homomorphism : I→ R defined as (a) Show that is a ring homomorphism. (b) Use FIT for rings to show that I/JR as rings. ([*]) = x - y.arrow_forwardLet R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a homomorphismarrow_forward
- Is the map y: C → C defined by y(x + iy) = x? = y? a ring isomorphism of C? Is it a ring homomorphism?arrow_forward4. Let y : R → S be a ring homomorphism. Prove that o' : R[X] → S[X] given by o'(ao + a1X + ..anX") = 4(ao) + y(a1)X + ... + p(am)X" is a ring homomorphism.arrow_forwardRing.Hom.arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,Linear Algebra: A Modern IntroductionAlgebraISBN:9781285463247Author:David PoolePublisher:Cengage Learning