����ر齬 I ���� No.7

���� 7.1 (������1)   ${\mbox{${\mathbb{Z}}$}}/(565,355)$ �ˤ����� $n\in {\mbox{${\mathbb{Z}}$}}$ �Υ��饹�� $\bar{n}$ �Ƚñ¤¯¤ï¿½ï¿½È¤Ë¤ï¿½ï¿½ë¡£ ���ΤȤ���

1. $\overline{ 565}=\bar{0}$ �Ǥ��뤳�Ȥ򼨤��ʤ�����
2. $\overline{ 355}=\bar{0}$ �Ǥ��뤳�Ȥ򼨤��ʤ�����
3. $\overline{5}=\bar{0}$ �Ǥ��뤳�Ȥ򼨤��ʤ�����

���� 7.2 (��1)   ${\mathbb{C}}[X]/(X^2-49,X^2-15 X +56)$ �ˤ����� $p\in {\mathbb{C}}[X]$ �Υ��饹�� $\bar{p}$ �Ƚñ¤¯¤ï¿½ï¿½È¤Ë¤ï¿½ï¿½ë¡£ï¿½ï¿½ï¿½Î¤È¤ï¿½ï¿½ï¿½

1. $\overline{ X^2-49}=\bar{0}$ , ���� $\overline{ X^2-15 X +56}=\bar{0}$ �Ǥ��뤳�Ȥ򼨤��ʤ�����
2. $\overline{X-7}=\bar{0}$ �Ǥ��뤳�Ȥ򼨤��ʤ�����
3. $\bar{X^5}$ ��Ǥ��������ñ�ʷ��˽񤭤ʤ�����
4. Ǥ�դ� $p\in {\mathbb{C}}[X]$ ���Ф��� $\overline {p(X)}=\overline{p(7)}$ ������Ω�Ĥ��Ȥ򼨤��ʤ�����

���� 7.3 (��1)   ${\mathbb{C}}[X,Y]/(X-5,Y-7)$ �ˤ����� $p\in {\mathbb{C}}[X,Y]$ �Υ��饹�� $\bar{p}$ �Ƚñ¤¯¤ï¿½ï¿½È¤Ë¤ï¿½ï¿½ë¡£ï¿½ï¿½ï¿½Î¤È¤ï¿½ï¿½ï¿½

1. $\overline{ X}=\overline{5}$ , ���� $\overline{ Y}=\bar{7}$ �Ǥ��뤳�Ȥ򼨤��ʤ�����
2. $\overline{X^2+3 Y +7 XY}$ ���ñ�ʷ���ľ���ʤ�����
3. Ǥ�դ� $p\in {\mathbb{C}}[X,Y]$ ���Ф��� $\overline {p(X,Y)}=\overline{p(5,7)}$ ������Ω�Ĥ��Ȥ򼨤��ʤ�����

��� 7.1   $R$ ��ñ�̸�����IJĴ��ġ�$I$ �ò¤½¤Î¥ï¿½ï¿½Ç¥ï¿½ï¿½ï¿½È¤ï¿½ï¿½Þ¤ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½Î¤È¤ï¿½ï¿½ï¿½

1. $I$ �� $R$ ���ǥ��ǥ���Ǥ���Ȥ����Τϡ� �� $f,g \in A, fg\in I$ �ʤ�С�$f$ �� $g$ �ΰ����� $I$ �θ��Ǥ���פ�����Ω�ĤȤ��˸����ޤ���
2. $I$ �� $R$ �ζ��祤�ǥ���Ǥ���ȸ����Τϡ�$I$ ����ʬ����Ȥ��ƴޤ� $R$ �Υ��ǥ��뤬 $R$ �� $I$ ���Ȱʳ��ˤ�¸�ߤ��ʤ��Ȥ��˸����ޤ���

���� 7.4�� �Ǥϡ��ִġפȤ�����ñ�̸�����IJĴ��Ĥ�ؤ����Ȥˤ��ޤ���

���� 7.4   �� $R$ �Υ��ǥ��� $I$ ���ǥ��ǥ���Ǥ��뤿���ɬ�׽�ʬ���ϡ� $R/I$ ������Ǥ��뤳�ȤʤΤò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.5   ${\mbox{${\mathbb{Z}}$}}$ �Υ��ǥ��� $n{\mbox{${\mathbb{Z}}$}}$ ���ǥ��ǥ���ˤʤ�Τϡ�$n=0$ ���� $n$ ���ǿ��ΤȤ��Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.6   �� $R$ �Υ��ǥ��� $I$ ��$R$ ������ $h$ ���� $Rh+I=R$ ���������ʤ�С�$h$ �ξ�;�� $\bar{h}$ �� $R/I$ �βĵո��Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.7   �� $R$ �ζ��祤�ǥ��� $I$ ��Ϳ����줿�Ȥ��� $R/I$ ���ΤǤ��뤳�Ȥò¡¢¼ï¿½ï¿½Î½ï¿½ï¿½ï¿½Ç¼ï¿½ï¿½ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

1. $R$ ��Ǥ�դθ� $h$ ���Ф��ơ�$Rh+I$ �� $I$ ����ʬ����Ȥ��ƴޤ।�ǥ���Ǥ��롣
2. $Rh+I=I\quad {\Leftrightarrow}\quad h\in I$ .
3. $R/I$ �ˤ����ơ� $\bar{h}\neq 0$ �ʤ�� $\bar{h}$ �� $R/I$ �βĵո��Ǥ��롣

���� 7.8   ���祤�ǥ�����ǥ��ǥ���Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.9  

1. ${\mbox{${\mathbb{Z}}$}}$ �Υ��ǥ��� $n{\mbox{${\mathbb{Z}}$}}$ , $m{\mbox{${\mathbb{Z}}$}}$ ($n,m$ ����������)�ˤĤ��ơ�
   $$n{\mbox{${\mathbb{Z}}$}}\supset m{\mbox{${\mathbb{Z}}$}}\quad {\Leftrightarrow}\quad \text{$n$ ¤Ï $m$ ¤ÎÇÜ¿ô¤Ç¤¢¤ë}$$
   �򼨤��ʤ�����
2. ${\mbox{${\mathbb{Z}}$}}$ �ζ��祤�ǥ���򤹤٤Ƶ��ʤ�����

���� 7.10   ${\mbox{${\mathbb{Z}}$}}[X]$ �Υ��ǥ��� $I=(X)$ ���ǥ��ǥ���Ǥ��äơ����祤�ǥ���ǤϤʤ����Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.11   ���� ${\mbox{${\mathbb{Z}}$}}[\sqrt{2}]$ �Υ��ǥ�����ǥ��ǥ��뤫�ɤ��������ʤ�����

1. $I_1=(5)$
2. $I_2=(7)$

���� 7.12  

1. ${\mathbb{C}}[X]$ �Υ��ǥ��� $(X^2-49,X^2-15 X +56)$ �� ${\mathbb{C}}[X]$ �ζ��祤�ǥ��� �Ǥ��뤳�Ȥ򼨤��ʤ�����
2. ${\mathbb{C}}[X,Y]$ �Υ��ǥ��� $(X-5,Y-7)$ �� ${\mathbb{C}}[X,Y]$ �ζ��祤�ǥ���� ���뤳�Ȥ򼨤��ʤ�����

���� 7.13   �Ĥδ֤ν�Ʊ������ $f:R\to S$ ��Ϳ����줿�Ȥ��ޤ������λ� $I$ �� $S$ ���ǥ��ǥ���ʤ顢$f^{-1}(I)$ �� $R$ ���ǥ��ǥ���Ǥ��뤳�Ȥò¼¨¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

���� 7.14   ���ξ�����­���� $f,R,S,I$ �ò¸«¤Ä¤ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

1. $f:R\to S$ �ϴĤδ֤ν�Ʊ��������
2. $I$ �� $S$ �ζ��祤�ǥ��롣
3. $f^{-1}(I)$ �� $R$ �ζ��祤�ǥ���Ǥʤ���

(�ҥ��:0 ���Τζ��祤�ǥ���Ǥ���)

���� 7.15   $R={\mbox{${\mathbb{Z}}$}}/6{\mbox{${\mathbb{Z}}$}}$ ���¿�༰�� $R[X]$ �θ� $f,g$ �ǡ�����2����Ʊ������­�����Τ���ʤ�����

1. $\deg(f)=\deg(g)=2$ .
2. $\deg(fg)=3 (<\deg(f)+\deg(g))$ .

���� 7.16   $R$ ��ġ�$I$ �ò¤½¤Î¥ï¿½ï¿½Ç¥ï¿½ï¿½ï¿½È¤ï¿½ï¿½Þ¤ï¿½ï¿½ï¿½ï¿½ï¿½$R/I$ �Υ��ǥ�������Τȡ�$I$ ����ʬ����Ȥ��ƴޤ� $R$ �Υ��ǥ�������ΤȤδ֤ˤϰ��а��б����Ĥ����Ȥò¼¡¤Î¤è¤¦ï¿½Ë¤ï¿½ï¿½Æ¼ï¿½ï¿½ï¿½ï¿½Ê¤ï¿½ï¿½ï¿½ï¿½ï¿½

1. $J$ �� $I$ ��ޤ� $R$ �Υ��ǥ���Ȥ���Ȥ���
   $$\bar{J}=\{\bar{x}; x\in J\}$$   ($?$ �� $?$ �Υ��饹��ɽ����)
   �� $R/I$ �Υ��ǥ���Ǥ��롣
2. $K$ �� $R/J$ �Υ��ǥ���Ȥ���Ȥ���
   $$\hat{K}=\{x\in R; \bar{x}\in K\}$$
   �� $R$ �Υ��ǥ���Ǥ��롣
3. �����Ĥ��б��ϸߤ���¾�ε��б��ˤʤäƤ��롣���ʤ����
   $$\hat{\bar{J}}=J,\quad \bar{\hat{K}}=K.$$