Results 1 to 3 of 3

Math Help - Abstract algebra: commutative ring proof

  1. #1
    Junior Member
    Joined
    Oct 2011
    Posts
    28

    Cool Abstract algebra: commutative ring proof

    Herstein Abstract Algebra. Ch 4. Sect 3. Problem 1.

    If R is a commutative ring and e ∈ R, let L(a) = {x ∈ R | xa = 0}. Prove that L(a) is an ideal of R
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member Tinyboss's Avatar
    Joined
    Jul 2008
    Posts
    433

    Re: Abstract algebra: commutative ring proof

    What do you normally check to see whether a subset of a ring is an ideal?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2011
    Posts
    28

    Re: Abstract algebra: commutative ring proof

    0 = (0)a ∈ L(a),
    so L(a) is non-empty.

    x₁,x₂ ∈ L(a)
    ⇒ x₁a = x₂a = 0
    ⇒ (x₁ + x₂)a = x₁a + x₂a = 0,
    so L(a) is closed under addition.

    r ∈ R, x ∈ L(a)
    ⇒ (rx)a = r(xa) = r(0) = 0
    ⇒ rx ∈ L(a),
    so L(a) is closed under multiplication by elements of R.

    → L(a) is an ideal of R.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 0
    Last Post: April 23rd 2010, 11:37 PM
  2. abstract algebra proof
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: March 3rd 2010, 07:23 AM
  3. Abstract algebra proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: February 19th 2010, 05:23 AM
  4. abstract algebra proof
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 27th 2010, 02:47 PM
  5. Replies: 3
    Last Post: December 29th 2009, 04:13 PM

Search Tags


/mathhelpforum @mathhelpforum