Show their is only one ring morphism from Zn to Zk if k|n

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In summary, the conversation discusses the proof that the function f: Zn to Zk, defined as [x]n ---> [x]k for all x in Zn, is a ring morphism if k|n. This is shown by verifying the properties and it is then questioned if this is the only ring morphism from Zn to Zk. It is suggested to start by assuming there is another ring morphism and using contradiction, or to directly show that f(1)=1.
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Daveyboy
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Homework Statement


For the rings Zn and Zk show that if k|n, then the function f: Zn to Zk
s.t [x]n --->[x]k for all x in Zn
is a ring morphism. Show this is the only ring morphism from Zn to Zk.

The attempt at a solution

So I showed it is a ring morphism by just verifying the properties, no big deal. I have no idea how to show that it is the only one though.

I want to start out by contradiction and assume that their is another one... but I don't know what to do with that. Is there a way to show it directly?
 
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  • #2
Once you know what f(1) is, you pretty much know what the morphism is, right? Is there any choice but to make f(1)=1?
 

1. What is a ring morphism?

A ring morphism is a function between two rings that preserves the ring structure. This means that it maps the ring elements and operations of one ring to the corresponding elements and operations in the other ring.

2. Why is there only one ring morphism from Zn to Zk if k|n?

This is because the ring Zn has unique factorization, meaning that every element in Zn can be expressed as a product of prime powers. Since k|n, the prime factors of k will also be factors of n, thus any ring morphism from Zn to Zk must preserve this factorization and there can only be one possible mapping.

3. How does the divisibility of k and n affect the number of ring morphisms?

If k and n are relatively prime (meaning they have no common factors), then there can be multiple ring morphisms from Zn to Zk. However, if k|n, then there is only one possible ring morphism as mentioned in the previous answer.

4. Can there be multiple ring morphisms if k does not divide n?

Yes, there can be multiple ring morphisms from Zn to Zk if k does not divide n. This is because in this case, the prime factors of k do not necessarily have to be factors of n, allowing for more flexibility in the mapping.

5. How does the concept of a ring morphism relate to other mathematical concepts?

Ring morphisms are closely related to other mathematical concepts such as group homomorphisms and vector space linear transformations. They all involve the preservation of structures and operations between two mathematical objects.

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