Ideal of definition - Knowlpedia

Definition

Let $A$ be a commutative topological ring. An open ideal $I\subseteq A$ is an ideal of definition if

I,\ I^2,\ I^3,\ldots

is a neighborhood basis of $0$ . Equivalently, the given topology on $A$ is the $I$ -adic topology.

In the classical adic convention used for formal schemes, an ideal of definition is required to be finitely generated. A topological ring that has such an ideal is an adic ring.

Choice of ideal

An ideal of definition is not generally unique. If $I$ and $J$ are ideals of definition for the same topology, their powers are cofinal: for every $m$ there is an $n$ with $I^n\subseteq J^m$ , and conversely. Thus the topology and its completion do not depend on which ideal of definition is chosen.

For the discrete topology, $0$ is an ideal of definition. A nonzero nilpotent ideal can also define the discrete topology.

References

The Stacks Project Authors, “Topological rings and modules.” Section 15.36, Tag 07E8.