An ideal with unit. So I'm trying to show that $A=R$ if $A$ is an ideal of a ring $R$ and has a unit.
I want to say this:
Let $a \in I$ a unit. And we know that $a^{-1} \in R,$ so $a  a^{-1} = 1 \in I$
And then I'll show that if $1 \in A$ then $A=R.$ (I have this covered already.)
My issue is that I don't know if I can assure that $a^{-1} \in R,$ I think that probably will be true, but I want to be sure about it, can you please help me noticing that?

Being of unit of $R$ means being invertible in $R$, so that $a^{-1}$ is in $R$ indeed. In what else could it be, btw ?
