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
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 $\displaystyle 3) \Rightarrow 1)$.