nLab adic noetherian ring

Idea

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

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

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

Related concepts

• completion of a ring

• linear topological ring

• pro-ring

References

• PlanetMath I-adic topology

• Wikipedia, Krull topology