Show that if \(\operatorname{Spec}(R) \subseteq \operatorname{MaxSpec}(R)\) and \(R\) is an integral domain, then \(R\) is a UFD.
Solution
Because \(R\) is an integral domain, the zero ideal \((0)\) is prime. Hence \[ (0)\in\operatorname{Spec}(R). \] By hypothesis every prime ideal is maximal, so \((0)\) is maximal. Therefore \[ R/(0)\cong R \] is a field.
Every field is a UFD. Thus \[ \boxed{R\text{ is a UFD}.} \]