Results 1 to 3 of 3

Math Help - Herstein: Subrings and Ideals

  1. #1
    Junior Member
    Joined
    Oct 2011
    Posts
    28

    Cool Herstein: Subrings and Ideals

    Herstein's Abstract Algebra: Ch 4. Sect 3. # 5.

    If I is an ideal of R and A is a subring of R, show that I ∩ A is an ideal of A
    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

    Re: Herstein: Subrings and Ideals

    Quote Originally Posted by ThatPinkSock View Post
    Herstein's Abstract Algebra: Ch 4. Sect 3. # 5.

    If I is an ideal of R and A is a subring of R, show that I ∩ A is an ideal of A
    So where exactly are you stuck?

    To show a set is an ideal you need to varify two things.

    1. The set is a sub group under "+"

    2. That the set "absorbs" elements under multipication.
    For any element r in the ring and any elment x in the ideal that
    xr is in the ideal and rx is in the ideal.

    Try to show the above and post back if you get stuck.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2011
    Posts
    28

    Cool Re: Herstein: Subrings and Ideals

    (1) [ I ∩ A, +] is a group because [I, +] and [A, +] are subgroups of R and the intersection of two subgroups is a subgroup.

    (2) Let x ∈ I ∩ A and let r ∈ A. Then x ∈ I and x ∈ A. Since I is an ideal of R, xr ∈ I and rx ∈ I. Since x ∈ A, xr ∈ A and rx ∈ A. So xr ∈ I ∩ A and rx ∈ I ∩ A.

    From (1) and (2) I ∩ A is an ideal of A.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Product ideals vs. products of ideals
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 17th 2011, 05:24 AM
  2. Prime Ideals, Maximal Ideals
    Posted in the Advanced Algebra Forum
    Replies: 8
    Last Post: March 7th 2011, 07:02 AM
  3. Commutative Ring, Subgroups, Subrings, Ideals
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 6th 2009, 03:56 AM
  4. Subrings and Ideals Questions
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: April 23rd 2009, 11:11 PM
  5. When are principal ideals prime ideals?
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: December 5th 2008, 12:18 PM

Search Tags


/mathhelpforum @mathhelpforum