There are no ring homomorphisms from Z5 to Z7

[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.

 

[StatusX]

Then the first condition, that 1 maps to 1, isn't satisfied.

 

[JaysFan31]

Could you just explain why it isn't satisfied? I think I'm missing something.

 

[StatusX]

You're saying f(1)=0, and 0 is not 1 in Z7

 

[JaysFan31]

Why does f(1)=0?

 

[matt grime]

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.

 

[JaysFan31]

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.

 

[JaysFan31]

What I'm basically asking is, is there a ring homomorphism from Z5 to Z7. My book says no. Why is this?

 

[StatusX]

If f(1)=1, then what is f(1+1), f(1+1+1), etc.? Eventually there will be a problem.

 

[matt grime]

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?

 

[JaysFan31]

Yeah what's the problem?
The identity requirement seems to hold. I'm really missing something. Could you spell it out for me?

 

[StatusX]

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?

 

[JaysFan31]

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.

 

[JaysFan31]

Can someone just update me on this?

 

[StatusX]

Keep going. What is 5 in Z5? In Z7?

 

[JaysFan31]

OK. I think I got it. Thanks for the help.