1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Rings and Homomorphism example

  1. Mar 11, 2009 #1
    1. The problem statement, all variables and given/known data
    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?


    2. Relevant 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


    3. 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 =)
     
  2. jcsd
  3. Mar 11, 2009 #2
    can the ring be a division ring?
     
  4. Mar 11, 2009 #3
    I don't think so, because I don't think we've learned about division rings yet lol
     
  5. Mar 11, 2009 #4

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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!
     
  6. Mar 11, 2009 #5
    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.
     
  7. Mar 11, 2009 #6

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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*
     
  8. Mar 11, 2009 #7
    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.
     
  9. Mar 11, 2009 #8

    Hurkyl

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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...)
     
  10. Mar 11, 2009 #9
    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.
     
  11. Mar 12, 2009 #10

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    What kind of functions do you know that turn addition into multiplication (or vice versa)? That's what the question is asking you.
     
  12. Mar 12, 2009 #11
    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!
     
  13. Mar 12, 2009 #12

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Rings and Homomorphism example
  1. Ring homomorphism (Replies: 3)

  2. Ring homomorphism (Replies: 10)

  3. Ring homomorphism (Replies: 1)

Loading...