Results 1 to 6 of 6

Math Help - Center as an ideal....

  1. #1
    Newbie
    Joined
    Nov 2009
    Posts
    9

    Center as an ideal....

    Hello all. How can I show that the center of a ring Z(R) is an ideal of a ring R iff R is commutative?

    Thanks!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Junior Member guildmage's Avatar
    Joined
    Aug 2009
    From
    Philippines
    Posts
    35
    Why don't you start with the easier part of the proof. Start with "If R is commutative, then Z(R) is an ideal in R."
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2009
    Posts
    9
    Well, the center is every element in the ring that commutes with the whole ring. So if the whole ring is commutative, the center will be the whole ring. So Z(R)=R, so Z(R) is an ideal....the other way of the if and only if proof is what I have been struggling with...
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Apr 2009
    Posts
    678
    Thanks
    1
    Quote Originally Posted by smacktalk88 View Post
    Well, the center is every element in the ring that commutes with the whole ring. So if the whole ring is commutative, the center will be the whole ring. So Z(R)=R, so Z(R) is an ideal....the other way of the if and only if proof is what I have been struggling with...
    Hint - Under the assumption Z(R) is ideal - show for every 'r' in R; 'r' belongs to Z(R)

    This is not all that tough, I guess. Just use the definition.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2009
    Posts
    9
    Ahhhh. Thanks a lot. This should help.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Newbie
    Joined
    Dec 2009
    Posts
    2
    Quote Originally Posted by smacktalk88 View Post
    Hello all. How can I show that the center of a ring Z(R) is an ideal of a ring R iff R is commutative?

    Thanks!

    If I\subseteq R is an ideal and 1\in I then obviously I=R ... apply this to I= Z(R)
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ideal,nil,nilpotent ideal in prime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 24th 2011, 08:57 AM
  2. Center of a Ring and an Ideal
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: May 14th 2011, 03:38 PM
  3. prove N is a maximal ideal iff N is a prime ideal
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 20th 2011, 10:02 AM
  4. Rings: when the center is an ideal
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: December 14th 2009, 03:58 PM
  5. Prime ideal but not maximal ideal
    Posted in the Advanced Algebra Forum
    Replies: 3
    Last Post: November 14th 2007, 10:50 AM

Search Tags


/mathhelpforum @mathhelpforum