Rings and Homomorphism example

[dancergirlie]

Homework Statement

Give an example of a ring R and a function f: R---->R such that f(a+b)=f(a)f(b) for all a,b in R. and f(a) is not the zero element for all a in R. Is your function a homomorphism?

Homework Equations

Let R and S be rings. A function f:R----->S is said to be a homomorphism if
f(a+b)=f(a) + f(b) and f(ab)=f(a)f(b) for all a,b in R

The Attempt at a Solution

Not really sure where to start here,
I was thinking about using Zn as my ring, perhaps with n as a prime number, so that way f(a) wouldn't be zero for any a. but i don't know what my function would be to satisfy that. Any help would be greatly appreciated =)

 

[JamesF]

can the ring be a division ring?

 

[dancergirlie]

I don't think so, because I don't think we've learned about division rings yet lol

 

[Hurkyl]

but i don't know what my function would be to satisfy that.

By "that", do you mean f(a+b)=f(a)f(b)?

That's an easy problem -- if you want f to satisfy that, and if you don't know a particular value of f, then the equation tells you how to compute it!

 

[dancergirlie]

I don't think that is what I need to do. It asks to give a specific ring and function that fits those specifications. Like for ring Z,
f(x)=x^2, but that doesn't work.

 

[Hurkyl]

If you're going to look for a ring and a function for which f(a+b)=f(a)f(b) is satisfied, you might as well use that equation to help you figure out what f should be. *shrug*

 

[dancergirlie]

I'm not sure if I follow you here, should I just plug in values from the ring and see if I can notice a pattern? I am just completely lost with this problem, I don't even know which ring I should use.

 

[Hurkyl]

Even if you don't notice a pattern, it will help you get started narrowing down a few specific values of f.

What ring to use? You had a few ideas you wanted to try, right? Do those! Or... you could start with values that every ring has. (e.g. 0, 1, 2...)

 

[dancergirlie]

I seriously have been looking at this for at least an hour now and have made no progress... I have tried the ring of even numbers, just Z, the ring Z5. I just don't know what to do, I'm not exactly an expert at the whole rings thing yet.

 

[matt grime]

What kind of functions do you know that turn addition into multiplication (or vice versa)? That's what the question is asking you.

 

[dancergirlie]

Thank you so much! I didn't even think of the exponential functions. I got that my function was obviously not homomorphic since f(a+b) did not equal f(a)+f(b) and f(ab) was not equal to f(a)f(b). I think I was just too tired at the moment to think of that function. Thanks for the help everyone!

 

[matt grime]

That isn't quite the right reasoning for why such a map cannot be an homomorphism. Why can't f(a)f(b) equal f(a)+f(b)? Just because the expressions look different doesn't mean that they are, really. Just take the possibly illegal case of the ring with one element :0.

But the question is trivial since it provides a reason why f can't be a homomorphism in its own statement: you are told that f(a) is never 0, and homomorphisms send 0 to 0. Note that the question precludes the example of the ring with one element from being considered.