adic noetherian ring in nLab

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Contents

Idea

A topological ring RR 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 I nI^n (n>0n\gt 0) where I⊂RI\subset R is a fixed ideal of RR, and RR 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 RR. If RR is an adic noetherian ring, an ideal J⊂RJ\subset R is a defining ideal iff it is open and its powers tend to {0}\{0\}.

The topology of an adic noetherian ring RR with the defining ideal II is said to be the II-adic topology and the descending filtration of RR by the powers of II to be the II-adic filtration.

For an adic noetherian ring RR there is a construction of a ringed space, its formal spectrum Spf(R)Spf(R), which does not depend on the choice of the ideal I⊂RI\subset R generating its (fixed in advance) topology. The underlying topological space of Spf(R)Spf(R) is Spec(R/I)Spec(R/I) which is (under the above assumptions on RR and II) a closed subspace of the spectrum Spec(R)Spec(R) and it contains all closed points of Spec(R)Spec(R).

• completion of a ring

• linear topological ring

• pro-ring

References

• PlanetMath I-adic topology

• Wikipedia, Krull topology