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 $T\cdot V:=\{t\cdot v\mid t\in T,v\in V\}\subset U$ holds. An element $a\in A$ is called power-bounded, if the set $\{a^{n}\mid n\in \mathbb {N} \}$ is bounded.

Examples

• An element $x\in \mathbb {R}$ 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 $\mathbb {Q} _{p}$, the p-adic numbers, is $\mathbb {Q} _{p}^{\circ }=\mathbb {Z} _{p}$.
• Every topologically nilpotent element is power-bounded.

Literature

• Morel: Adic spaces

• Wedhorn: Adic spaces