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 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.Examples
- An element is power-bounded if and only if hold.
- 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, denoted by. This follows from the ultrametric inequality.
- The ring of power-bounded elements in is.
- Every topological nilpotent element is power-bounded.
Literature
- Morel:
- Wedhorn: