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
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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.
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.
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