Null Sequences form Maximal Left and Right Ideal/Lemma 2

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.

Then:

$\NN$ is a maximal left ideal.

Proof

By Lemma 1 of Null Sequences form Maximal Left and Right Ideal:

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

Hence $\NN$ is a left ideal of $\CC$.

It remains to show that $\NN$ is maximal.

Lemma 7

$\NN \subsetneq \CC$.

$\Box$

Lemma 8

There is no left ideal $\JJ$ of $\CC$ such that $\NN \subsetneq \JJ \subsetneq \CC$

$\Box$

The result follows by definition of maximal left ideal.

$\blacksquare$

Sources

• 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions