3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphismYn: 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 imageof , and use the First Isomorphism Theorem to deduce thatZ/(6) ≈ Z₂ X Z3.

Here given that ring homomorphism , Here for each is onto .Now another defined map is Here we have…

3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphism Pn Z→ Zn taking a to the remainder on dividing a by n. Show that the map : : ZZ2 x Z3, taking a € Z to (p2(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.

3. We have seen in lectures that, for any integer n > 0, there is a ring homomorphism Pn Z→ Zn taking a to the remainder on dividing a by n. Show that the map : : ZZ2 x Z3, taking a € Z to (p2(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.

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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Question

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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Number 6: show at K is splitting over Q.