Rings - Ideals - Equivalent statements? - Math Help Forum

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#1

$Old$ December 15th, 2008, 11:10 AM

chrisneedshelp $chrisneedshelp is offline$

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$Default$ 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

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$Old$ December 15th, 2008, 12:34 PM

clic-clac $clic-clac is offline$

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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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