Results 1 to 2 of 2

Math Help - Rings

  1. #1
    Member
    Joined
    Oct 2008
    Posts
    91

    Rings

    Suppose that R is a set with addition and multiplication defined that satisfy all of the axioms of a ring except possibly for a+b=b+a (commutativity of addition). Prove that a+b=b+a, and hence that R is a ring.

    Now, we want to show that the group (R, +) is abelian. Since, not every group is abelian, we will have to use the properties of multiplication. The only axiom that links addition to multiplication is the axiom of distributivity so if my reasoning is correct the proof should use that axiom. However, I am a little stuck on this and would really appreciate any hints.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    May 2008
    Posts
    2,295
    Thanks
    7
    Quote Originally Posted by nmatthies1 View Post
    Suppose that R is a set with addition and multiplication defined that satisfy all of the axioms of a ring except possibly for a+b=b+a (commutativity of addition). Prove that a+b=b+a, and hence that R is a ring.

    Now, we want to show that the group (R, +) is abelian. Since, not every group is abelian, we will have to use the properties of multiplication. The only axiom that links addition to multiplication is the axiom of distributivity so if my reasoning is correct the proof should use that axiom. However, I am a little stuck on this and would really appreciate any hints.
    we do need R to have 1, the multiplicative identity, otherwsie your claim would be wrong. the proof now is quite easy:

    (1+1) \cdot (a+b)=1 \cdot (a+b) + 1 \cdot (a+b)=a+b+a+b and (1+1) \cdot (a+b)=(1+1) \cdot a + (1+1) \cdot b=a+a+b+b. thus a+b+a+b=a+a+b+b and the result follows.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring theory, graded rings and noetherian rings
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: January 4th 2012, 11:46 AM
  2. a book on semigroup rings and group rings
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: October 2nd 2011, 04:35 AM
  3. Rings.
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 23rd 2010, 11:54 PM
  4. rings
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: January 21st 2010, 07:33 AM
  5. Rings with 0 = 1.
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 26th 2009, 05:34 AM

Search Tags


/mathhelpforum @mathhelpforum