If is a commutative noetherian ring, every localization of the form where is also noetherian. If a scheme is locally noetherian this implies that there is a base of topology of consisting of spectra of noetherian rings. In particular every affine subscheme in has a basis of topology of spectra of noetherian rings.
An affine scheme is a spectrum of a noetherian ring iff it is a locally noetherian scheme.
Every affine subscheme of a locally noetherian scheme is the spectrum of a noetherian ring. A scheme is noetherian if and only if it has a finite cover by spectra of noetherian rings.