Results 1 to 2 of 2

Math Help - Ring Homomorphism Questions

  1. #1
    Member
    Joined
    Nov 2008
    Posts
    152

    Ring Homomorphism Questions

    1) Consider the mapping p: Z3 --> Z6 given by p(x) = 2x, for x = 0,1,2. Is p a ring homomorphism? How about the mapping p(x) = remainder of 4x (mod 6)?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by Janu42 View Post
    1) Consider the mapping p: Z3 --> Z6 given by p(x) = 2x, for x = 0,1,2. Is p a ring homomorphism? How about the mapping p(x) = remainder of 4x (mod 6)?
    You first need to consider if [x]_3\mapsto [2x]_6 is well-defined. Say that [x]_3 = [y]_3 then x\equiv y(\bmod 3), so 2x\equiv 2y(\bmod 6) and hence [2x]_6 = [2y]_6 which means the mapping is well-defined. Now what about [x]_3\mapsto [4x]_3? Try to argue that this is also well-defined. To show it is a ring homorphism (depending on how you define "ring homomorphism") you just need to show f(x+y) = f(x)f(y) and f(xy) = f(x)f(y). Notice f( [x]_3) + f([y]_3)= [2x]_6 + [2y]_6 = [2(x+y)]_6 = f([x+y]_3). Also, f([x]_3)f([y]_3) = [2x]_6\cdot [2y]_6 = [4xy]_6 \not = [2xy]_6 = f([xy]_3). Thus, it is not a homorphism, you try doing the second case.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 10th 2011, 07:49 AM
  2. Ring Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: February 27th 2011, 09:14 PM
  3. Ring Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: November 1st 2010, 12:53 AM
  4. homomorphism ring ~
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 23rd 2009, 10:34 AM
  5. Ring Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 1st 2008, 03:01 PM

Search Tags


/mathhelpforum @mathhelpforum