Results 1 to 3 of 3

Math Help - Is this a ring homomorphism?

  1. #1
    Junior Member
    Joined
    Oct 2008
    Posts
    38

    Is this a ring homomorphism?

    Hi, could you please help with the following question. I need to understand the method in proving it's a homomorphism and then i can do other similar questions i've been asked to complete.

    Let Z x Z be the ring which is the cartesian product of two copies of Z. Determine if the following is a ring homomorphism.

    ZxZ -> Z, h(x,y)=x

    (Z is the set of integers, and -> is an arrow)

    Thanks for you help.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Behold, the power of SARDINES!
    TheEmptySet's Avatar
    Joined
    Feb 2008
    From
    Yuma, AZ, USA
    Posts
    3,764
    Thanks
    78
    Quote Originally Posted by Louise View Post
    Hi, could you please help with the following question. I need to understand the method in proving it's a homomorphism and then i can do other similar questions i've been asked to complete.

    Let Z x Z be the ring which is the cartesian product of two copies of Z. Determine if the following is a ring homomorphism.

    ZxZ -> Z, h(x,y)=x

    (Z is the set of integers, and -> is an arrow)

    Thanks for you help.
    Remember to be a homomorphism it must preserve addition and multiplication.

    let (a,b) \in \mathbb{Z} \times \mathbb{Z} and (c,d) \in \mathbb{Z} \times \mathbb{Z}

    h[(a,b)+(c,d)]=h[(a+c,b+d)]=a+c=h[(a,b)]+h[(c,d)]

    Since you didn't define mult on ordered pairs I'm not sure about your ring ops, But it would be verified in the same manner.

    I hope this helps.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2008
    Posts
    38

    To show: the set of complex numbers of the form a+bi where a,b in Q is isomorphic to

    Thanks so much for your help.

    I have another question on rings and fields;

    Let F be the set of all complex numbers of the form a+bi with a, b in Q. (Q= rationals) Show that F is a subring of C (C=complex numbers), and that it is a field. Show that F is isomprphic to the field of fractions of the ring Z[i] of Gaussian integers.


    I have shown that F is a subring of the complex numbers using the subring test.

    Is it correct to prove F is a field by showing it has a one (1), is not the zero ring, every non-zero number has an inverse (a/(a^2 +b^2) -(b/(a^2 +b^2))i) and that it is commutative (a+bi) +(c+di) = (c+di) + (a+bi)?

    Finally is showing that F is isomorphic to the field of fractions of the ring Z[i] of Gaussian integers the same as showing that the Guassian numbers are isomorphic to the field of Gaussian rationals.

    How do i show that these are isomorphic?

    Thanks a lot for your help.
    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: 1
    Last Post: August 7th 2010, 08:22 AM
  3. homomorphism ring ~
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 23rd 2009, 10:34 AM
  4. Ring homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 12th 2009, 04:50 AM
  5. Ring Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 1st 2008, 03:01 PM

Search Tags


/mathhelpforum @mathhelpforum