Null Sequences form Maximal Left and Right Ideal/Corollary 1

Theorem

Let $\struct {F, \norm {\, \cdot \,} }$ be a valued field.

Let $\CC$ be the ring of Cauchy sequences over $F$.

Let $\NN$ be the set of null sequences in $F$.

That is:

$\NN = \set {\sequence {x_n}: \ds \lim_{n \mathop \to \infty} x_n = 0}$

Then $\NN$ is a maximal ring ideal of $\CC$.

Proof

By Null Sequences form Maximal Left and Right Ideal then $\NN$ is a maximal left ideal of $\CC$.

A field is by definition a commutative ring.

In a commutative ring, a maximal left ideal is by definition a maximal ideal.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions
• 2007: Svetlana Katok: p-adic Analysis Compared with Real: $\S 1.3$ Construction of the completion of a normed field