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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- Dec 15th 2008, 09:10 AMchrisneedshelpRings - 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 - Dec 15th 2008, 10:34 AMclic-clac
I guess that A means I :)

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

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

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