Principal ideal domains

$\mathbb{Z}$. Shown earlier: every ideal is $n\mathbb{Z}$ for some $n\ge0$.

$\mathbb{R}[X]$. Let $I$ be a nonzero ideal. Let $P$ be a nonzero element of $I$ of minimal degree. For any $Q\in I$, Euclidean division: $Q=AP+R$ with $\deg R<\deg P$. Since $R=Q-AP\in I$ and $\deg R<\deg P$, by minimality $R=0$. So $Q=AP\in(P)$. Hence $I=(P)$.

Conclusion. Both are PIDs. Since every Euclidean domain is a PID, and $\mathbb{Z}$ and $\mathbb{R}[X]$ are Euclidean, this is consistent.