Past exam of the commutative algebra course of the University of Oxford / 2023 / 1 / e / i

Because $(r)$ is a proper prime ideal, each $(r^k)$ is proper and its radical of an ideal isIndeed, $(r^k)\subseteq (r)$ gives one inclusion of radicals of ideals, while $(rc{)}^k\in (r^k)$ gives the other.

We prove the saturation condition for a primary ideal: if $ab\in (r^k)$ and $a\notin \mathfrak{p}$, then $b\in (r^k)$. For $k=1$ this is the defining property of a prime ideal. For $k>1$, write $ab=r^{k}c$. Since $a\notin (r)$, the prime ideal condition gives $b=r{b}_1$. The element $r$ is a non-zero-divisor, so cancellation givesInduction gives $b_1\in (r^{k-1})$, hence $b\in (r^k)$. Together with the computed radical of an ideal, this proves