1. Let Z6¹Z₂ × Z3 be defined by the following specification of its values 0 + (0,0), 1↔X(1, 1), 2 (0, 2), 3 (1,0), 4 (0, 1), 5 → (1,2) Use the addition and multiplicationtables of Z6 and Z₂ x Z6 to show that f is a ring isomorphism.

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Answered: 1. Let Z6 Z2 x Z3 be defined by the…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

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If f: Z→ Z is the map defined by f(x)=2x, Is f a ring homomorphism when Z has its usual ring operations? How would you prove that? If not, what could be a counter-example?
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3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphism Yn: Z → Zn taking a to the remainder on dividing a by n. Show that the map : Z→ Z2 × Z3, taking a € Z to (42(a), 3(a)), is a ring homomorphism. Find the kernel and image of , and use the First Isomorphism Theorem to deduce that Z/(6) ≈ Z₂ X Z3.
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1. Let Z6 Z2 x Z3 be defined by the following specification of its values 0 → (0, 0), 1 → (1,1), 2 → (0,2), 3(1,0), 4(0,1), 5(1, 2) Use the addition and multiplication tables of Z6 and Z₂ x Z6 to show that f is a ring isomorphism.