I need to prove that there exist no nontrivial homomorphisms from Z to.docx
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The document demonstrates that there are no nontrivial homomorphisms from the integers (Z) to Z. It proves that any ring homomorphism must map 1 to either 0 or 1, resulting in trivial homomorphisms. Consequently, the only possible homomorphisms are f(x) = 0 and f(x) = x.
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![I need to prove that there exist no nontrivial homomorphisms from Z to Z. ( Trivial
homomorphisms being f(x) = 0 and f(x) = x )
Solution
If f is a ring homomorphism from Z to Z then
f(1) = f(1*1) = f(1) * f(1)
So we have
0 = f(1) * f(1) - f(1)
0 = f(1) * [f(1) - 1]
Since Z has no zero divisors, we must have
f(1) = 0 or f(1) = 1.
In the former case f is the trivial homomorphism f(x) = 0. In the latter, f is the trivial
homomorphism f(x) = x, since it is fairly easy to convince yourself that for any x in Z we have](https://image.slidesharecdn.com/ineedtoprovethatthereexistnonontrivialhomomorphismsfromzto-230128091555-0b6b9b0f/85/I-need-to-prove-that-there-exist-no-nontrivial-homomorphisms-from-Z-to-docx-1-320.jpg)
I need to prove that there exist no nontrivial homomorphisms from Z to.docx
- 1. I need to prove that there exist no nontrivial homomorphisms from Z to Z. ( Trivial homomorphisms being f(x) = 0 and f(x) = x ) Solution If f is a ring homomorphism from Z to Z then f(1) = f(1*1) = f(1) * f(1) So we have 0 = f(1) * f(1) - f(1) 0 = f(1) * [f(1) - 1] Since Z has no zero divisors, we must have f(1) = 0 or f(1) = 1. In the former case f is the trivial homomorphism f(x) = 0. In the latter, f is the trivial homomorphism f(x) = x, since it is fairly easy to convince yourself that for any x in Z we have
- 2. f(x) = x * f(1).