Results 1 to 5 of 5

Math Help - Ring help

  1. #1
    Junior Member
    Joined
    Mar 2007
    Posts
    40

    Ring help

    Two questions:
    1. If R is a ring then is every element of R either a unit or zero-divisor.

    COnversely

    2. IF R is a ring, then there is no elements of R which is both a unit and a zero-divisor.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by chadlyter View Post
    Two questions:
    1. If R is a ring then is every element of R either a unit or zero-divisor.
    Is R a finite ring?

    Because, if it is then we can prove it as follows:

    If x is a zero-divisor the proof is complete.

    If x is not a zero-divisor then consider the finite set,
    {a_1x,a_2x,...,a_nx} where a_i are the non-zero elements of the ring (assuming it is non-trivial).
    Then, we can show none are equal to each other.
    (Excercise).
    Then by Dirichlet's Pigeonhole Principle, is an enumeration of all non-zero elements in R. Hence we can find an a_i such that a_i x = 1. Thus, x is a unit.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by TheNextDirichlet
    Is R a finite ring?
    It has to be.
    Because, for example, consider, Z.
    The integers under addition and multiplication.

    The element 2 in Z, it neither a unit nor zero divisor
    (However, for all finite rings this is true).
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Mar 2007
    Posts
    40

    RIng

    The Ring is not defined as a finite ring. These are true/false questions I have encountered in a abstract algebra class. If false I have to give a counterexample. For example, If R is a ring, then every element of R is either a unit or zero-divisor. In my mind this is false but the counterexample is what is messing with my head.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by chadlyter View Post
    If false I have to give a counterexample. For example,
    It is false consdier the example I gave you.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 1
    Last Post: October 23rd 2011, 06:36 AM
  2. [SOLVED] Prove the Artinian ring R is a division ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: June 8th 2011, 03:53 AM
  3. example of prime ring and semiprime ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 27th 2011, 05:23 PM
  4. Ideals of ring and isomorphic ring :)
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 24th 2009, 03:23 AM
  5. ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: July 27th 2009, 09:07 PM

Search Tags


/mathhelpforum @mathhelpforum