Let $R$ be the number ring related to a field $K$ of finite degree over $\mathbb Q$, i.e. $\mathbb Q\le K\le\mathbb C$ and $[K:\mathbb Q]=n$. Hence $R=\mathbb A\cap K$, where $\mathbb A$ is the ring of the algebraic integers in $\mathbb C$.
Suppose that $\forall \;I\unlhd R\;\;\exists\; a\in R$ s.t. $aI=R$.
Now in my book I read that from this follows that every ideal is principal. But i think found a (really) stronger fact: if $aI=R$, being $R=(1)$, then there exists $b\in I$ s.t. $ab=1$. But $ab\in I$, hence $I$ must be $R$.
Am I right?
Thank you all