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Math Help - Rings, and zero divisors

  1. #1
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    Rings, and zero divisors

    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.

    a. show that R has no divisors of 0
    b. show that bab = b
    c. show that R has unity
    d. show that R is a division ring


    B I worked out but I'm not sure if it works..
    we're given aba = a and we need bab=b
    abab= ab
    multiply by inverse of a to get bab=b

    thanks!
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  2. #2
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    Quote Originally Posted by lttlbbygurl View Post

    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.

    a. show that R has no divisors of 0.
    suppose xy=0 but x \neq 0 and y \neq 0. so there exists a unique z \in R such that xzx=x. but then: x(z+y)x=xzx + xyx=x. therefore z+y=z, and hence y=0. contradiction.

    b. show that bab = b.
    we have baba=ba and thus (bab - b)a=0. since a \neq 0, the first part of the problem gives us: bab-b=0.

    c. show that R has unity.
    let 0 \neq a \in R. so there exists b \in R such that aba=a. the claim is that ab=1_R. so let c \in R. then caba=ca and hence (cab - c)a=0. therefore cab=c, by the first part of the problem.

    also by the second part, bab=b and thus babc=bc, which gives us b(abc - c)=0. thus abc = c.

    d. show that R is a division ring.
    let 0 \neq x. so there exists y \in R such that xyx=x and thus (xy - 1)x=0, which gives us: xy=1.
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