Null Sequences form Maximal Left and Right Ideal/Lemma 3
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 right ideal.
Proof
By Lemma 1 of Null Sequences form Maximal Left and Right Ideal then $\NN$ is an ideal of $\CC$.
Hence $\NN$ is a right ideal of $\CC$.
It remains to show that $\NN$ is maximal.
By Lemma 7 of Null Sequences form Maximal Left and Right Ideal then $\NN \subsetneq \CC$.
By maximal right ideal it needs to be shown that:
- There is no right ideal $\JJ$ of $\CC$ such that $\NN \subsetneq \JJ \subsetneq \CC$
Let $\JJ$ be a Right ideal of $\CC$ such that $\NN \subsetneq \JJ \subseteq \CC$.
It will be shown that $\JJ$ = $\CC$, from which the result will follow.
Let $\sequence {x_n} \in \JJ \setminus \NN$
By Inverse Rule for Cauchy sequences then
- $\exists K \in \N: \sequence {\paren {x_{K + n} }^{-1} }_{n \mathop \in \N}$ is a Cauchy sequence.
Let $\sequence {y_n}$ be the sequence defined by:
- $y_n = \begin{cases} 0 & : n \le K \\ \paren {x_n}^{-1} & : n > K \end{cases}$
By Cauchy Sequence with Finite Elements Prepended is Cauchy Sequence then $\sequence {y_n} \in \CC$
By the definition of a right ideal the product $\sequence {x_n} \sequence {y_n} = \sequence {x_n y_n} \in \JJ$
By the definition of $\sequence {y_n}$ then:
- $x_n y_n = \begin{cases} 0 & : n \le K \\ 1 & : n > K \end{cases}$
Let $\mathcal 1 = \tuple {1, 1, 1, \dots}$ be the unity of $\CC$
Then $\mathcal 1 - \sequence {x_n} \sequence {y_n}$ is the sequence $\sequence {w_n}$ defined by:
- $w_n = \begin {cases} 1 & : n \le K \\ 0 & : n > K \end {cases}$
By Convergent Sequence with Finite Elements Prepended is Convergent Sequence then $\sequence {w_n}$ is convergent to 0.
So $\sequence {w_n} \in \NN \subsetneq \JJ$
Since \sequence {x_n} $\sequence {y_n}, \sequence {w_n} \in \JJ$ by the definition of a ring ideal then:
- $\sequence {w_n} + \sequence {x_n} \sequence {y_n} = \mathcal 1 \in \JJ$
By the definition of a right ideal then:
- $\forall \sequence {a_n} \in \CC, \mathcal 1 \circ \sequence {a_n} = \sequence {a_n} \in \JJ$
Hence $\JJ = \CC$
$\blacksquare$
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction: $\S 3.2$: Completions