Classify all radical ideals in the ring of integers \(\mathbb{Z}\).
Solution
Every ideal of \(\mathbb Z\) is \((n)\) for a unique \(n\ge0\).
The zero ideal is radical because \(\mathbb Z\) is reduced. Now let \[ n=p_1^{a_1}\cdots p_r^{a_r} \] with distinct primes \(p_i\). Put \[ \operatorname{rad}(n)=p_1\cdots p_r. \] Then \[ \sqrt{(n)}=(\operatorname{rad}(n)). \] Indeed, if \(x^m\in(n)\), then every \(p_i\) divides \(x\), so \(\operatorname{rad}(n)\mid x\). Conversely, if \(\operatorname{rad}(n)\mid x\), then for \[ N\ge\max_i a_i \] we have \(n\mid x^N\).
Thus \((n)\) is radical exactly when \[ n=\operatorname{rad}(n), \] i.e. when \(n\) is squarefree. Including \(n=1\) and \(n=0\), the radical ideals are precisely \[ \boxed{(0)\text{ and }(p_1\cdots p_r)\text{ with the }p_i\text{ distinct}}. \]