 Principal ideal criterion for a nontrivial rank-one valuation ring

A valuation ring for a nontrivial real-valued rank-one valuation is a principal ideal domain exactly when its value group is discrete. A least positive value makes the value group cyclic and lets every nonzero ideal be generated by an element of smallest valuation. Conversely, if the maximal ideal is principal, its generator has the least positive value. Nontriviality matters: a field with the trivial valuation is a principal ideal domain, but not a discrete valuation ring in the usual nontrivial convention.