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 be a topological ring. A subset is called bounded, if, for every neighbourhood of zero, there exists an open neighbourhood of zero such that holds. An element is called power-bounded, if the set is bounded.Wedhorn: Def. 5.27
Examples
An element is power-bounded if and only if . More generally, if is a topological commutative ring whose topology is induced by an absolute value, then an element is power-bounded if and only if holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring (a valuation ring), denoted by . This follows from the ultrametric inequality. The ring of power-bounded elements in , the p-adic numbers, is . Every topologically nilpotent element is power-bounded.Wedhorn: Rem. 5.28 (4)
Literature
Morel: Adic spaces
Wedhorn: Adic spaces
References