A power-bounded element is an element of a

topological ring In mathematics, a topological ring is a ring R that is also a topological space such that both the addition and the multiplication are continuous as maps: R \times R \to R where R \times R carries the product topology. That means R is an additive ...

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 A neighbourhood (Commonwealth English) or neighborhood (American English) is a geographically localized community within a larger town, city, suburb or rural area, sometimes consisting of a single street and the buildings lining it. Neighbourh ...

U

of zero, there exists an open neighbourhood

V

of zero such that

T \cdot V := \ \subset U

holds. An element

a \in A

is called power-bounded, if the set

\

is bounded.

Examples

* An element

x \in \mathbb R

is power-bounded if and only if

, x,  \leq 1

hold. * More generally, if

A

is a topological commutative ring whose topology is induced by an

absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if x is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), ...

, 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 In mathematics, a subring of a ring is a subset of that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and that shares the same multiplicative identity as .In general, not all s ...

, denoted by

A^

. This follows from the ultrametric inequality. * The ring of power-bounded elements in

\mathbb Q_p

\mathbb Q_p^{\circ} = \mathbb Z_p

. * Every topological nilpotent element is power-bounded.Wedhorn: Rem. 5.28 (4)

Literature

* Morel
Adic spaces
* Wedhorn
Adic spaces

References

Topological algebra