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

{\displaystyle A}

be a topological ring. A subset

T

⊂

A

{\displaystyle T\subset A}

is called bounded, if, for every neighbourhood

U

{\displaystyle U}

of zero, there exists an open neighbourhood

V

{\displaystyle V}

of zero such that

T

⋅

V

:=

{

t

⋅

v

∣

t

∈

T

,

v

∈

V

}

⊂

U

{\displaystyle T\cdot V:=\{t\cdot v\mid t\in T,v\in V\}\subset U}

holds. An element

a

∈

A

{\displaystyle a\in A}

is called power-bounded, if the set

{

a

n

∣

n

∈

N

}

{\displaystyle \{a^{n}\mid n\in \mathbb {N} \}}

is bounded.

Examples

An element

x

∈

R

{\displaystyle x\in \mathbb {R} }

is power-bounded if and only if

|

x

|

≤

1

{\displaystyle |x|\leq 1}

.

More generally, if

A

{\displaystyle A}

is a topological commutative ring whose topology is induced by an absolute value, then an element

x

∈

A

{\displaystyle x\in A}

is power-bounded if and only if

|

x

|

≤

1

{\displaystyle |x|\leq 1}

holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring (a valuation ring), denoted by

A

∘

{\displaystyle A^{\circ }}

. This follows from the ultrametric inequality.

The ring of power-bounded elements in

Q

p

{\displaystyle \mathbb {Q} _{p}}

, the p-adic numbers, is

Q

p

∘

=

Z

p

{\displaystyle \mathbb {Q} _{p}^{\circ }=\mathbb {Z} _{p}}

.