There are no ring homomorphisms from Z5 to Z7

In summary, the conversation discusses a problem in algebra class that requires proving that there are no ring homomorphisms from Z5 to Z7. The definition of ring homomorphism is provided and the conversation explores the contradiction in the definition and why the function f(x)=0 does not work as a homomorphism. The conversation also discusses the elements 1, 1+1, 1+1+1, etc. in Z5 and how they map to Z7, ultimately concluding that there is no ring homomorphism from Z5 to Z7.
  • #1
JaysFan31
I just need confirmation.

I have a problem in my algebra class that says:
Prove that there are no ring homomorphisms from Z5 to Z7.
I have the following definition of ring homomorphism:
Let R and S be rings. A function R to S is a ring homomorphism if the following holds:
f(1R)=1S.
f(r1+r2)=f(r1)+f(r2) for all r1 and r2 in R.
f(r1r2)=f(r1)f(r2) for all r1 and r2 in R.

I've been thinking and wouldn't f(x)=0 work?
This is a problem in a published textbook so it doesn't make sense to me. Could anyone clue me into where there might be a contradiction in the definition?

Thanks in anticipation. Mike.
 
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  • #2
Then the first condition, that 1 maps to 1, isn't satisfied.
 
  • #3
Could you just explain why it isn't satisfied? I think I'm missing something.
 
  • #4
You're saying f(1)=0, and 0 is not 1 in Z7
 
  • #5
Why does f(1)=0?
 
  • #6
Because you said it did. You asked: why is the map f(x)=0 for all x not a homomorphism. Ans: because f(1) is not 1, contradicting the definition of ring homomorphism.
 
  • #7
Well the f(x)=0 wasn't part of the problem. It was just my own thinking. Does this still work? Somehow I'm still not getting where there is a contradiction in the definition.
 
  • #8
What I'm basically asking is, is there a ring homomorphism from Z5 to Z7. My book says no. Why is this?
 
  • #9
If f(1)=1, then what is f(1+1), f(1+1+1), etc.? Eventually there will be a problem.
 
  • #10
JaysFan31 said:
Well the f(x)=0 wasn't part of the problem. It was just my own thinking. Does this still work?


Does what still work?
 
  • #11
Yeah what's the problem?
The identity requirement seems to hold. I'm really missing something. Could you spell it out for me?
 
  • #12
So what are the elements 1, 1+1, 1+1+1, ... in Z5? Are any of them the same? If so, do they map to the same element in Z7, as they must?
 
  • #13
Are you saying that this function is injective and therefore not a ring homomorphism?
Because I don't see how 3 in Z5 not being the same as 3 in Z7 is a reason for it not being a homomorphism.
 
  • #14
Can someone just update me on this?
 
  • #15
Keep going. What is 5 in Z5? In Z7?
 
  • #16
OK. I think I got it. Thanks for the help.
 

1. Why can't there be any ring homomorphisms from Z5 to Z7?

Ring homomorphisms are functions that preserve the algebraic structure of a ring. Since Z5 and Z7 have different algebraic structures, it is not possible for a function to preserve both structures. Therefore, there cannot be any ring homomorphisms from Z5 to Z7.

2. What is the algebraic structure of Z5 and Z7?

Z5 and Z7 are both rings, which means they have two operations (addition and multiplication) that follow certain rules, such as associativity and distributivity. However, Z5 has 5 elements while Z7 has 7 elements, so they have different underlying structures.

3. Can there be any other types of homomorphisms between Z5 and Z7?

Yes, there can be other types of homomorphisms between Z5 and Z7, such as group homomorphisms or field homomorphisms. These homomorphisms preserve different structures, such as the group or field structure, but not the ring structure.

4. Is it possible to find a function that maps Z5 to Z7 without preserving the ring structure?

Yes, it is possible to find a function that maps Z5 to Z7 without preserving the ring structure. However, this function would not be considered a ring homomorphism, as it does not preserve the algebraic structure of a ring.

5. How do we prove that there are no ring homomorphisms from Z5 to Z7?

To prove that there are no ring homomorphisms from Z5 to Z7, we can use a proof by contradiction. We assume that there exists a ring homomorphism from Z5 to Z7, and then show that this leads to a contradiction, thereby proving that our assumption was incorrect.

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