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Math Help - Prove that if every element in a ring R except 1 has a left quasi-inverse, then R is

  1. #1
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    Prove that if every element in a ring R except 1 has a left quasi-inverse, then R is

    Hello !
    can you help me proving this exercise please :

    An element a in a ring R has a left quasi inverse if there exist an element b in R with a+b-ab=0.
    Prove that if every element in a ring R except 1 has a left quasi-inverse, then R is
    a division ring..

    THANK YOU !
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  2. #2
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    Re: Prove that if every element in a ring R except 1 has a left quasi-inverse, then R

    In other words, show that if every element, other than 1, has a left quasi-inverse, then every element, other than 0, has an inverse.
    I take it we are allowed to assume that the ring has a multiplicative identity, 1?
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  3. #3
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    Re: Prove that if every element in a ring R except 1 has a left quasi-inverse, then R

    here is something to get you started:

    a + b - ab = 0

    a + (1 - a)b = 0

    1 + (1 - a)b = 1 - a

    1 = (1 - a) - (1 - a)b

    1 = (1 - a)(1 - b).

    so 1 - a has a right-inverse, unless a = 1. why does this show U(R) = R\{0}?
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  4. #4
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    Re: Prove that if every element in a ring R except 1 has a left quasi-inverse, then R

    Thank you very much for your help
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