��II�� No.10

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��Ϳ��Ƥ��Ȥ��롣

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�� $m_1,m_2,\dots, m_k$ ��Ƥ��ơ��餬 �ط��

�� 10.1 �Ĥ��Τ褦�ˤ��ơ��δ֤δط�� (�ط� R) ��ߤ��褦�ʲ÷� $M^{(R)}$ �Ȥ�� $\{m_j^{(R)}\}_{j=1}^k$ ��Ĥ��롣

$F=A^{\oplus k}$ ��ͤ��褦�� Ĥθ�(�ִ��ܥ٥��ȥ��) $e_1,e_2,\dots, e_k$ ��롣
$v_i=\sum_{j=1}^k r_{i j} e_j$ �Ȥ�� ʬ�÷� �ǡ� $\{v_i\}_{i=1}^l$ ��Τ� �ͤ��褦�� -��ΤǤ��Ȥ��äƤ�褤��
��;�÷� $M^{(R)}=F/N$ ��ͤ��褦�� θ��ֺ�� 뤫�ɤ�� ǥ��饹ʬ��饹��ΤǤ��롣
$M^{(R)}=F/N$ �Ǥ� �Υ��饹�� $m_j^{(R)}$ �Ȥ��ȡ� �� $\{m_j^{(R)}\}$ ��졢 ��ϴط��

$% latex2html id marker 1153 $\displaystyle \sum_{j} r_{ij} m_j^{(R)}=0 \qquad (i=1,2,\dots,l)$$ (�ط� $R^{(R)}$ )

((�ط� R) �� $m_j^{(R)}$ ��Ѥ��)��ߤ��
��餫�ʤ褦�ˡ� �ط� $R^{(R)}$ �� $\{m_j^{(R)}\}$ �� ��ܴط���Ǥ��롣 ��ʤ��[�� 7.1 �� $\phi$ �ȸƤФ�Ƥ��Τˤ��] -��Ʊ��

$\displaystyle A^{\oplus k} \ni \begin{pmatrix} a_1\\ a_2 \\ \vdots \\ a_k \end{pmatrix}\mapsto \sum _j a_j m_j^{(R)} \in M^{(R)}$

�γˤ� $\{v_i\}_{i=1}^l$ ��롣

̿�� 10.2

-�÷� �� Ĥθ� $m_1,\dots, m_k$ ��ơ��ϴط�� (R) ��ߤ��Ȥ��롣 $M^{(R)}, \{m_j^{(R)}\}$ ��嵭�Τ褦�˹��Ȥ�� -�÷��ν�Ʊ��

$\displaystyle \varphi: M^{(R)} \to M$

�� $% latex2html id marker 1192 $ \varphi(m_j^{(R)})=m_j \qquad(j=1,2,\dots, k)$$ ��Τ��ҤȤ� ¸�ߤ��롣�� $\varphi$ ��ɬ��Ū��ͤǤ��롣

��ͤ��ǥ��ˤʤ뤳�Ȥ�¿�� 㤨�С��̿��褦�� $(M^{(R)}, \{ m_j^{(R)}\})$ �� (Ŭ��ʰ�̣��)Ʊ��ư�դǤ��뤳�Ȥ��Ǥ��롣

�� 10.3 $A={\mbox{${\mathbb{Z}}$}}$ ,

�Ȥ��

$% latex2html id marker 1207 $\displaystyle 2 m_1 + 3 m_2 =0,\quad 6 m_1 +6 m_2=0$$

(Rexample)

�ʤ�ط��÷�

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�Ȥ��ƤϤ��Ȥ��С�0 -�÷� (

)�ʤɤȤ��ʤ�Τ⤢�뤷��Ǥʤ��Ƥ� $% latex2html id marker 1216 $ m_1=0,\quad 3 m_2=0$$ �ʤ�� ( $M={\mbox{${\mathbb{Z}}$}}/3{\mbox{${\mathbb{Z}}$}}, m_1=0, m_2=[1]_3$ ) �� 롣��֡��礭�ʡ�(��Ū��)��Τ�

$\displaystyle M^{(R)}={\mbox{${\mathbb{Z}}$}}^{\oplus 2}/ ({\mbox{${\mathbb{Z}}$}}.(2 e_1 +3 e_2)+ {\mbox{${\mathbb{Z}}$}}.(6 e_1 +6 e_2)$

�Ȥ��ơ� $m_1^{(R)},m_2^{(R)}$ �򤽤줾��

�Υ��饹�Ƿ�᤿��ΤǤ��롣 (�� $(M^{(R)},m_1^{(R)},m_2^{(R)})$ �Ϻ��ξ��ˤ� $({\mbox{${\mathbb{Z}}$}}/2 {\mbox{${\mathbb{Z}}$}}\oplus {\mbox{${\mathbb{Z}}$... ...b{Z}}$}},\begin{pmatrix}1\\ 0\end{pmatrix}, \begin{pmatrix}0\\ 1 \end{pmatrix})$ ��Ʊ��Ǥ��롣)

PID ��ͭ��÷�

�ι�¤��̿��7.7�Τ褦�ˤ褯�狼�äƤ��ơ� �Ȥ��

��ɬ��ͭ��ɽ��Ǥ��ä��

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��Ϳ��줿�Ȥ�� ;�÷�

��Ф��̿��7.7 ��Ѥ��뤳�Ȥˤ�ꡢ��Τ��Ȥ��狼�롣

��̿��ϡ� �͡��ľ��Ǥ�դ�ͭ��÷��Ǥ�դ��ʬ�÷��Ϥޤ�ͭ��Ǥ��롢 �Ȥ��̤ξ��Ǥ��롣

�Ĵ� PID �� ͭ��÷��ι�¤�ϡ��ñ��פȤ��Τ�� Τ�ɽ΢��Τδط��ˤ��롣��ϼ��ͳ�ˤ��: ̿�� 10.2 �ˤ��ȡ� ��̤ˡ��

�θ��ʬ�˻��Ĺ��

�� 10.1 ${\mbox{${\mathbb{Z}}$}}$ -�÷� $F={\mbox{${\mathbb{Z}}$}}\oplus {\mbox{${\mathbb{Z}}$}}$ �ˡ��饹ʬ�� (`` $\sim$ '')��

$% latex2html id marker 1288 $\displaystyle m_1 \sim m_2 \qquad {\Leftrightarro... ... 4 \end{pmatrix}+ {\mbox{${\mathbb{Z}}$}} \begin{pmatrix} 9 \\ 8 \end{pmatrix}$$

�� $m\in F$ �Υ��饹��

��ɽ��Ȥˤ��롣��ΤȤ��

$% latex2html id marker 1294 $\displaystyle v_1= \begin{pmatrix} 1 \\ 1 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 1 \\ 5 \end{pmatrix}$$

�Ȥ��ȡ� �Ǥ��뤳�Ȥ򼨤��ʤ��
�� $0\in F$ �Υ��饹 �Ȥ��ʤ��Ȥ򼨤��ʤ��

�����II���� No.10

��II�� No.10