1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Ring theory problem

  1. Jan 25, 2008 #1
    1. The problem statement, all variables and given/known data
    Let R be a ring that contains at least two elements. Suppose for each nonzero a in R, there exists a unique b in R such that aba=a.
    Show that R has no divisors of 0.

    2. Relevant equations

    3. The attempt at a solution
    Let a*c=0 where a,c are not equal to 0.
    aba=a implies aba-a=0=nac where n is any integer which implies that a(ba-1-nc)=0
    I am not seeing the contradiction.
  2. jcsd
  3. Jan 25, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    You are given that b is unique for each a. Show that if ac=0, aba=a then there is another value d not equal to b such that ada=a.
  4. Jan 25, 2008 #3
    Next part of the question: Show that R has unity.

    I need to show that there exists an element 1 of R such that 1a=a1=1 for all a in A.

    We can use cancellation now that we showed R has no divisors of 0, so bab=b.

    Let a_1, a_2 be nonzero. Then there exists b_1 and b_2 such that [itex]a_1 b_1 a_1 = a_1[/itex] and [itex]a_2 b_2 a_2 = a_2[/itex].

    To complete the proof, I need to show that [itex] a_1 b_1 = a_2 b_2 = b_1 a_1 = b_2 a_2 [/itex], right?

    I can show that [itex] b_1 a_1 = a_2 b_2 [/itex] from the fact that [itex] a_1 b_1 a_1 a_2 = a_1 a_2 = a_1 a_2 b_2 a_2 [/itex] and similarly I can show that [itex] a_1 b_1 = b_2 a_2 [/itex] but I am having trouble showing that the left and right identity are the same.
  5. Jan 27, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    Consider the set G of all elements of the ring such that x^2=x. The products you are talking about have that property. Now consider two elements such that x^2=x and y^2=y. So x^2*y=x*y^2. Now cancel your way down to x=y.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Ring theory problem
  1. Ring theory problem (Replies: 3)

  2. Ring theory problem (Replies: 1)

  3. Ring theory problem (Replies: 2)