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...
Related questions
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.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd0a2eb85-71bc-41a8-8f67-a7e3c7209cc3%2Fda480d84-ab97-4d9d-a045-1b43444c94e7%2F2mqr3tn_processed.jpeg&w=3840&q=75)
Transcribed Image Text: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.
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 4 steps with 32 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Follow-up Questions
Read through expert solutions to related follow-up questions below.
Recommended textbooks for you
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning
![Elements Of Modern Algebra](https://www.bartleby.com/isbn_cover_images/9781285463230/9781285463230_smallCoverImage.gif)
Elements Of Modern Algebra
Algebra
ISBN:
9781285463230
Author:
Gilbert, Linda, Jimmie
Publisher:
Cengage Learning,
![Linear Algebra: A Modern Introduction](https://www.bartleby.com/isbn_cover_images/9781285463247/9781285463247_smallCoverImage.gif)
Linear Algebra: A Modern Introduction
Algebra
ISBN:
9781285463247
Author:
David Poole
Publisher:
Cengage Learning