����ر齬 I ���� No.7

���� 7.1 (��Ʊ������)   $f:R\to S$ ��Ĥδ֤ν�Ʊ���Ȥ���Ȥ������Τ��Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

1. $f$ �γ� $I=\ker f=f^{-1}(0)$ �� $R$ �Υ��ǥ���Ǥ��롣
2. $f(r)=f(r') \quad {\Leftrightarrow}\quad r-r'\in I$ ��
3. $\bar{f}:R/I \ni \bar{r} \mapsto f(r) \in S$ ����ɽ���μ�꤫���ˤ�餺���������Ƥ��롣
4. $\bar{f}$ ��ñ�ͽ�Ʊ�������Ǥ��롣
5. $f$ �����ͤʤ�С�$\bar{f}$ ��Ʊ�������Ǥ��롣

���� 7.1 (��Ʊ�������λȤ���)  

(1).

$4{\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}$ �� ${\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}$ �Υ��ǥ���Ǥ��뤳�Ȥ򼨤��ʤ�����

(2).

$({\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}})/(4{\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}})$ ����衣

(1),(2)����פ˲�褹�롣�ʲ������� $m$ �ˤ������� $m$ �� ${\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}$ �Ǥ�Ʊ����� $[m]_{12}$ �Ƚ񤭡� $m$ �� ${\mbox{${\mathbb{Z}}$}}/3{\mbox{${\mathbb{Z}}$}}$ �Ǥ�Ʊ����� $[m]_4$ �Ƚ񤯤��Ȥˤ��롣 ���� $f:{\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}\to {\mbox{${\mathbb{Z}}$}}/4{\mbox{${\mathbb{Z}}$}}$ ��

$$f:[m]_{12} \to [m]_4$$

���������С����줬��ɽ���ˤ�餺�ˤ��ޤ��������Ƥ��ꡢ ��Ʊ�������Ǥ��뤳�ȤϤ�����ʬ���롣$f$ �γˤ�

$$\{\bar{m};\quad m$$    �� $4$ ���ܿ�$$\}=4{\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}$$

�ǡ���Ʊ�������γˤϥ��ǥ�������顢 $4{\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}$ �� ${\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}}$ �Υ��ǥ���Ǥ��롣 $(\to (1))$

$f$ �����ͤ����顢��Ʊ������(����)�ˤ�ꡢ $\bar{f}:({\mbox{${\mathbb{Z}}$}}/12{\mbox{${\mathbb{Z}}$}})/(4{\mbox{${\mathbb... ...12{\mbox{${\mathbb{Z}}$}}) \to {\mbox{${\mathbb{Z}}$}}/4{\mbox{${\mathbb{Z}}$}}$ ��Ʊ����Ϳ���롣 ($\to (2)$ (������ ${\mbox{${\mathbb{Z}}$}}/4{\mbox{${\mathbb{Z}}$}}$ ))

���� 7.2   ���δ� $R$ �Ȥ��Υ��ǥ��� $I$ �ˤĤ��ơ�$R/I$ ����衣

1. $R={\mbox{${\mathbb{Z}}$}}/48{\mbox{${\mathbb{Z}}$}},I=8R$ .
2. $R={\mathbb{C}}[X]/((X+1)(X-1){\mathbb{C}}[X]), I=(X+1)R$ .

���� 7.3   Ʊ�� ${\mathbb{C}}[X]/(X-3){\mathbb{C}}[X]\cong {\mathbb{C}}$ �ò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.4   ${\mbox{${\mathbb{Z}}$}}[\sqrt{-1}]/(1+\sqrt{-1}) \cong {\mbox{${\mathbb{Z}}$}}/2{\mbox{${\mathbb{Z}}$}}$ �ò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.5   $f:$$\mbox{${\mathbb{R}}$}$$\to S$ �� �Ĥν�Ʊ���Ȥ��ޤ��� $S$ �Υ��ǥ��� $J$ ��Ȥäơ�

$$I=f^{-1}(J)$$

�Ȥ����ȡ�$I$ �� $R$ �Υ��ǥ���Ǥ��äơ�ñ�ͽ�Ʊ�� $\bar{f}:R/I\to S/J$ �� $f$ �ˤ�꼫�����������뤳�Ȥ򼨤��ʤ�����

���� 7.6   $R={\mbox{${\mathbb{Z}}$}}[\sqrt{10}]$ �Υ��ǥ��� $I=(2,\sqrt{10})$ �ˤ���;�� $R/I$ �� ${\mbox{${\mathbb{Z}}$}}/2{\mbox{${\mathbb{Z}}$}}$ ��Ʊ���Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½ï¿½$I$ �� $R$ ���ǥ��ǥ���Ǥ��뤳�Ȥ�����ʤ�����