Null Sequences form Maximal Left and Right Ideal

Theorem

Let $\struct {R, \norm {\, \cdot \,} }$ be a normed division ring.

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

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

That is:

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

Then $\NN$ is a ring ideal of $\CC$ that is a maximal left ideal and a maximal right ideal.

Corollary

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

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

Proof

By every convergent sequence is a Cauchy sequence then $\NN \subseteq \CC$.

The proof is completed in these steps:

$(1): \quad \NN$ is an ideal of $\CC$.

$(2): \quad \NN$ is a maximal left ideal.

$(3): \quad \NN$ is a maximal right ideal.

Lemma 1

$\NN$ is an ideal of $\CC$.

$\Box$

Lemma 2

$\NN$ is a maximal left ideal.

$\Box$

Lemma 3

$\NN$ is a maximal right ideal.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.2$: Completions: Lemma $3.2.8$, Problem $80$
• 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.3$ Construction of the completion of a normed field