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Homework Help: 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

    Dick

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    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

    Dick

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    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.
     
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