# Rings - Ideals - Equivalent statements?

• December 15th 2008, 09:10 AM
chrisneedshelp
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
• December 15th 2008, 10:34 AM
clic-clac
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 $3) \Rightarrow 1)$.