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.

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 10E: Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove...

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Let :312 be defined by ([x]3)=4[x]12 using the same notational convention as in Exercise 9. Prove that is a ring homomorphism. Is (e)=e where e is the unity in 3 and e is the unity in 12?
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