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    Let  R be an integral domain and let  x, y \in R Show that  (x) = (y) if and only if  \exists \lambda \in R^* such that  x = \lambda y
    Where  (x) = \{rx: r \in R \}, R^* = \{x: x \ is \ a \ unit \} .
    Proving from left to right.
    In part of the answer it says  x \in (x)  = (y) \Rightarrow x = \lambda y  for some  \lambda \in R  and  y \in (y) = (x)  \Rightarrow y = \mu x for some  \mu \in R . Thus  x = \lambda y = \lambda \mu x \Rightarrow x-\lambda \mu x = 0_R \Rightarrow x(1_R - \lambda \mu) = 0_R \Rightarrow x=0_R or  (1_R - \lambda \mu) = 0_R .
    If  x = 0_R \Rightarrow y = \mu 0_R = 0_R \Rightarrow x = 1_R y . I don't understand the last implication, how does it imply  \lambda = 1_R from the case when  x= 0_R ?
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    If x=0 then it implies that y=0, and so x=1.y. In fact, for any \lambda it'd be true that x=\lambda y in this case; they just gave a concrete example of a value of \lambda that works.
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    Oh I see, thanks.

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Updated: April 13, 2012
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