a bAssume that the ring {: |a, b, c € Z \ is a ring with respect to matrix additionand multiplication and define the mapping o : R → Z by• ([: :) -a b= c.(a) Verify that the mapping ø is a homomorphism.(b) Verify that o is onto.(c) Find ker(ø).(d) Show that R/ ker ø is isomorphic to Z.(e) Is ker(ø) a prime ideal? Is it a maximal ideal?

Answered: Image /qna-images/answer/bb8ffc61-9294-420e-a11d-c3167964bf41.jpg

a b Assume that the ring { |a, b, c € Z} is a ring with respect to matrix addition and multiplication and define the mapping o : R → Z by •(::)-- a b = c. (a) Verify that the mapping o is a homomorphism. (b) Verify that ø is onto. (c) Find ker(6).

a b Assume that the ring { |a, b, c € Z} is a ring with respect to matrix addition and multiplication and define the mapping o : R → Z by •(::)-- a b = c. (a) Verify that the mapping o is a homomorphism. (b) Verify that ø is onto. (c) Find ker(6).

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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a b
Assume that the ring {: |a, b, c € Z \ is a ring with respect to matrix addition
and multiplication and define the mapping o : R → Z by
• ([: :) -
a b
= c.
(a) Verify that the mapping ø is a homomorphism.
(b) Verify that o is onto.
(c) Find ker(ø).
(d) Show that R/ ker ø is isomorphic to Z.
(e) Is ker(ø) a prime ideal? Is it a maximal ideal?

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