Results 1 to 3 of 3

Math Help - Ring isomorphism

  1. #1
    Junior Member
    Joined
    Nov 2008
    Posts
    50

    Ring isomorphism

    Hello I want to solve the following problem:

    Let R = Z( sqrt2 ) = { a+b sqrt2|a,b belong to Z.

    1. Prove that R is integral domain.

    2. If I = (  sqrt 2 ) is the ideal generated by sqrt(2), prove that R/I~=Z2 (~= means isomorphic).


    //////

    In 1. I showed that elements of R are commutative and without zero-divisors.


    In 2. I understood I as all elements of R multiplied by sqrt(2)) . Am I correct?

    How shall I prove isomorphism? I tried the following function from R to Z2: f(a+ sqrt2b+I)= a mod 2. But it is only hommomorphism in my opinion, not isomorphism.


    Can you help me?

    Thank you in advance!
    Last edited by andreas; November 16th 2008 at 04:38 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2008
    From
    Paris, France
    Posts
    1,174
    Quote Originally Posted by andreas View Post
    Hello I want to solve the following problem:

    Let R = Z( sqrt2 ) = { a+b sqrt2|a,b belong to Z.

    2. If I = (  sqrt 2 ) is the ideal generated by sqrt(2), prove that R/I~=Z2 (~= means isomorphic).

    In 2. I understood I as all elements of R multiplied by sqrt(2)) . Am I correct?

    How shall I prove isomorphism? I tried the following function from R to Z2: f(a+ sqrt2b+I)= a mod 2. But it is only hommomorphism in my opinion, not isomorphism.
    The mapping f:R\to\mathbb{Z}_2 defined by f(a+b\sqrt{2})=a\,({\rm mod}\,2) is an homomorphism, and it is obviously surjective. As a consequence, it suffices to show that its kernel is I to conclude that R/I\simeq \mathbb{Z}_2.
    Its kernel consists of numbers x=a+b\sqrt{2} where a,b\in\mathbb{Z} and a=2a' is even. Can you go on?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2008
    Posts
    50
    Thank you! It was very helpful!! I can go on
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: September 8th 2011, 03:55 PM
  2. Ring isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: October 12th 2010, 01:30 PM
  3. Isomorphism ring :P
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 23rd 2009, 10:39 AM
  4. Ring Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: November 2nd 2008, 02:57 AM
  5. Ring Isomorphism
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 27th 2008, 05:56 PM

Search Tags


/mathhelpforum @mathhelpforum