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Math Help - Rings - Ideals - Equivalent statements?

  1. #1
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    Rings - Ideals - Equivalent statements?

    I'm looking to prove that the following conditions are equivalent for an ideal I in a ring R.

    1) 1 is in A
    2) A contains a unit
    3) A = R


    thanks
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  2. #2
    Senior Member
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    I guess that A means I

    So since 1 is a unit,  1) \Rightarrow 2) is obvious.

    If I contains a unit u, then, by ideal's definition, \forall a \in R, (au^{-1})u=a \in I, so R \subset I. I\subset R because of ideal's definition. So 2) \Rightarrow 3).

    Finally, I=R \Rightarrow 1 \in R \subset I \Rightarrow 1\in I, therefore 3) \Rightarrow 1).
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