A topological ring is an adic noetherian ring if it is noetherian as a ring and it has a topological basis consisting of all translations of the neighborhoods of zero of the form () where is a fixed ideal of , and is Hausdorff and complete in that topology. A choice of such an ideal is said to be the defining ideal or (more French) the ideal of definition of the topological ring . If is an adic noetherian ring, an ideal is a defining ideal iff it is open and its powers tend to . The topology of an adic noetherian ring with the defining ideal is said to be the -adic topology and the descending filtration of by the powers of to be the -adic filtration.
For an adic noetherian ring there is a construction of a ringed space, its formal spectrum , which does not depend on the choice of the ideal generating its (fixed in advance) topology. The underlying topological space of is which is (under the above assumptions on and ) a closed subspace of the spectrum and it contains all closed points of .