Power-bounded element

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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.

Let ${\displaystyle A}$ be a topological ring. A subset ${\displaystyle T\subset A}$ is called bounded, if, for every neighbourhood ${\displaystyle U}$ of zero, there exists an open neighbourhood ${\displaystyle V}$ of zero such that ${\displaystyle T\cdot V:=\{t\cdot v\mid t\in T,v\in V\}\subset U}$ holds. An element ${\displaystyle a\in A}$ is called power-bounded, if the set ${\displaystyle \{a^{n}\mid n\in \mathbb {N} \}}$ is bounded.^[1]

• An element ${\displaystyle x\in \mathbb {R} }$ is power-bounded if and only if ${\displaystyle |x|\leq 1}$ hold.
• More generally, if ${\displaystyle A}$ is a topological commutative ring whose topology is induced by an absolute value, then an element ${\displaystyle x\in A}$ is power-bounded if and only if ${\displaystyle |x|\leq 1}$ holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring, denoted by ${\displaystyle A^{\circ }}$. This follows from the ultrametric inequality.
• The ring of power-bounded elements in ${\displaystyle \mathbb {Q} _{p}}$ is ${\displaystyle \mathbb {Q} _{p}^{\circ }=\mathbb {Z} _{p}}$.
• Every topological nilpotent element is power-bounded.^[2]

• Morel: Adic spaces

• Wedhorn: Adic spaces