鐃緒申鐃緒申惘藹� I 鐃緒申鐃緒申 No.8

$\fbox{準同型定理編(2)}$ 鐃緒申鐃緒申鐃�No.8鐃祝は￥申鐃瞬環」と醐申鐃緒申鐃緒申単鐃縮醐申鐃緒申鐃緒申腸調鐃緒申弔里鐃緒申箸鐃舜わ申鐃緒申鐃夙にわ申鐃殉わ申鐃緒申鐃殉わ申鐃緒申鐃瞬緒申同鐃緒申鐃竣わ申単鐃縮醐申鐃緒申鐃楯つわ申里鐃緒申鐃緒申鐃粛わ申鐃暑こ鐃夙にわ申鐃殉わ申鐃緒申

鐃緒申鐃緒申 8.1 $\mbox{${\mathbb{Q}}$}$ $% latex2html id marker 959 $ [\sqrt{2}]\cong$$ $\mbox{${\mathbb{Q}}$}$

$\mbox{${\mathbb{Q}}$}$

鐃薯示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.2 有鐃緒申鐃緒申鐃緒申 $\mbox{${\mathbb{Q}}$}$ 鐃緒申鐃緒申分鐃緒申鐃緒申 ${\mbox{${\mathbb{Z}}$}}[1/5]=\{m/5^n; m\in {\mbox{${\mathbb{Z}}$}}; n=0,1,2,\dots,)$ 鐃緒申 $\mbox{${\mathbb{Q}}$}$ 鐃緒申鐃緒申分鐃縦になるこ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.3

$\displaystyle (5X-1)$ $\displaystyle \mbox{${\mathbb{Q}}$}$ $\displaystyle [X]\cap {\mbox{${\mathbb{Z}}$}}[X]=(5X-1){\mbox{${\mathbb{Z}}$}}[X]$

鐃薯示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.4

$\displaystyle {\mbox{${\mathbb{Z}}$}}[1/5]\cong {\mbox{${\mathbb{Z}}$}}[X]/(5X-1){\mbox{${\mathbb{Z}}$}}[X]$

鐃薯示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.5 鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申

鐃緒申鐃術随申

鐃祝つわ申鐃銃￥申鐃緒申鐃緒申同鐃緒申鐃薯示わ申鐃淑わ申鐃緒申鐃緒申

$\displaystyle R[X]/IR[X] \cong (R/I)[X]$

鐃緒申鐃緒申 8.6 鐃緒申

鐃緒申鐃緒申分鐃緒申

鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申

鐃祝つわ申鐃銃￥申

$\displaystyle R+I=\{r+a;r\in R,a \in I\}$

鐃熟￥申

鐃緒申鐃緒申分鐃縦となるこ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.7 鐃緒申

鐃緒申鐃緒申分鐃緒申

鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申

鐃祝つわ申鐃銃￥申

$% latex2html id marker 1011 $\displaystyle S=R+I, \quad R\cap I=0 $$

鐃緒申鐃緒申鐃緒申立鐃銃ば￥申 $S/I\cong R$ 鐃夙なるこ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.8 鐃縦緒申同鐃緒申 $f:R\to S$ 鐃祝つわ申鐃銃￥申

鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申任鐃緒申鐃出￥申 $f^{-1}(J)$ 鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申箸覆襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 8.9 鐃縦緒申同鐃緒申 $f:R\to S$ 鐃祝つわ申鐃銃￥申

鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申里箸鐃緒申鐃�

鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申箸覆鐃殉わ申鐃緒申鐃緒申(鐃淑わ申覆鐃緒申鐃緒申反鐃緒申鐃藷げてわ申鐃緒申鐃緒申鐃緒申鐃緒申)

鐃緒申鐃緒申鐃粛なわ申匹鐃緒申任鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 8.10 ${\mbox{${\mathbb{Z}}$}}/7{\mbox{${\mathbb{Z}}$}}$ 鐃塾醐申鐃薯すべて緒申鐃渋べて￥申 $\bar{0}$ 鐃淑鰹申鐃塾醐申鐃塾逆醐申鐃薯それぞ鐃緒申鐃緒申覆鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 8.11

鐃緒申鐃塾とわ申鐃殉わ申鐃緒申鐃緒申鐃塾とわ申同鐃緒申 $K[X,Y]/XK[X,Y]\cong K[Y]$ 鐃薯示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.12 鐃緒申

鐃夙わ申鐃緒申鐃緒申分鐃緒申

鐃夙につわ申鐃銃￥申

鐃緒申

鐃緒申鐃叔ワ申鐃叔ワ申鐃緒申覆鐃出￥申 $P\cap R$ 鐃緒申

鐃緒申鐃叔ワ申鐃叔ワ申鐃緒申任鐃緒申襪鰹申箸鮗┐鐃緒申覆鐃緒申鐃緒申鐃緒申鐃緒申妊鐃緒申妊鐃緒申鐃竣わ申峩鐃緒申腑わ申妊鐃緒申鐃竣にわ申鐃緒申鐃緒申箸匹鐃緒申鐃緒申鐃�

鐃緒申鐃緒申 8.13 鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申鐃�

鐃祝つわ申鐃銃￥申 $I\cdot I$ 鐃緒申

鐃緒申略鐃緒申鐃緒申鐃殉わ申鐃緒申

鐃緒申

鐃塾ワ申鐃叔ワ申鐃緒申如鐃�

鐃夙なわ申弌鐃�

鐃夙なるこ鐃夙を示わ申鐃淑わ申鐃緒申鐃緒申

鐃緒申鐃緒申 8.14 鐃緒申鐃緒申

鐃塾醐申

鐃祝つわ申鐃銃￥申鐃緒申鐃薯示わ申鐃淑わ申鐃緒申鐃緒申

$a\vert b$ 鐃叔わ申鐃緒申(鐃緒申鐃淑わ申鐃緒申鐃緒申鐃緒申鐃� $c\in R$ 鐃緒申鐃緒申鐃獣て￥申鐃夙書けわ申)鐃緒申鐃夙と￥申 $aR\subset bR$ 鐃夙わ申同鐃粛でわ申鐃暑。
鐃緒申鐃夙わ申同伴鐃叔わ申鐃緒申(鐃緒申鐃淑わ申鐃緒申鐃� $a\vert b$ 鐃緒申鐃緒申 $b\vert a$ 鐃緒申鐃緒申鐃緒申立鐃緒申)鐃緒申鐃夙と￥申鐃夙わ申同鐃粛でわ申鐃暑。