Power-bounded element

A power-bounded element is an element of a topological ring whose powers are bounded. These elements are used in the theory of adic spaces.

Definition

Let $A$ be a topological ring. A subset $T \subset A$ is called bounded, if, for every neighbourhood $U$ of zero, there exists an open neighbourhood $V$ of zero such that $T \cdot V := {t \cdot v ∣ t \in T, v \in V} \subset U$ holds. An element $a \in A$ is called power-bounded, if the set ${a^{n} ∣ n \in ℕ}$ is bounded.

Examples

An element $x \in ℝ$ is power-bounded if and only if $| x | \leq 1$ .
More generally, if $A$ is a topological commutative ring whose topology is induced by an absolute value, then an element $x \in A$ is power-bounded if and only if $| x | \leq 1$ holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring (a valuation ring), denoted by $A^{\circ}$ . This follows from the ultrametric inequality.
The ring of power-bounded elements in $ℚ_{p}$ , the p-adic numbers, is $ℚ_{p}^{\circ} = ℤ_{p}$ .
Every topologically nilpotent element is power-bounded.

Literature

Morel: Adic spaces

Wedhorn: Adic spaces

References

↑ Wedhorn: Def. 5.27
↑ Wedhorn: Rem. 5.28 (4)

[1] Wedhorn: Def. 5.27

[2] Wedhorn: Rem. 5.28 (4)