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鐃緒申鐃緒申鐃� II 鐃緒申鐃緒申 No.6

鐃緒申鐃緒申鐃塾テ￥申鐃緒申:

\fbox{巾零元・根基イデアル/零因子・素イデアル}

鐃緒申鐃� 6.1   鐃緒申 $R$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦�$R/I$ 鐃緒申 $0$ 鐃淑鰹申鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申覆鐃緒申箸鐃緒申鐃� $I$ 鐃緒申 $R$ 鐃塾削申鐃署イデワ申鐃緒申任鐃緒申鐃夙醐申鐃緒申鐃緒申

鐃緒申鐃緒申 6.1   $k^n$ 鐃緒申任鐃春わ申鐃緒申鐃重�鐃緒申分鐃緒申鐃緒申 $V$ 鐃緒申鐃出わ申鐃銃￥申 鐃緒申鐃塾削申標鐃緒申 $A(V)$ 鐃緒申 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申覆鐃緒申鐃� 鐃緒申鐃緒申鐃緒申鐃獣て￥申$I(V)$ 鐃緒申多鐃準式鐃緒申 $k[X_1,\dots, X_n]$ 鐃塾削申鐃署イデワ申鐃緒申任鐃緒申襦�

鐃緒申鐃� 6.2   鐃緒申 $R$ 鐃塾ワ申鐃叔ワ申鐃緒申 $I$ 鐃叔￥申$R$ 鐃夙は異なわ申鐃塾わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃銃わ申鐃緒申箸鐃緒申襦� $R/I$ 鐃緒申 $0$ 鐃淑鰹申鐃緒申鐃緒申鐃緒申鐃述わ申鐃緒申鐃緒申覆鐃緒申箸鐃緒申鐃� $I$ 鐃緒申 $R$ 鐃緒申鐃叔ワ申鐃叔ワ申鐃緒申任鐃緒申鐃夙醐申鐃緒申鐃緒申

鐃緒申鐃� 6.3   $k^n$ 鐃塾駈申鐃叔なわ申鐃緒申鐃重�鐃緒申分鐃緒申鐃緒申 $V$ 鐃緒申鐃緒申鐃緒申任鐃緒申鐃夙は￥申

\begin{displaymath}V=V_1\cup V_2 , \quad V_1 \neq V , V_2 \neq V \end{displaymath}

鐃淑わ申鐃緒申鐃重�鐃緒申鐃緒申 $V_1,V_2$ 鐃緒申存鐃淳わ申鐃緒申箸鐃緒申砲鐃緒申鐃緒申鐃� $V$ 鐃緒申鐃緒申鐃緒申任覆鐃緒申箸鐃緒申法鐃�$V$ 鐃熟器申鐃緒申任鐃緒申鐃夙わ申鐃緒申鐃緒申

鐃緒申鐃� 6.4   $k^n$ 鐃塾駈申鐃叔なわ申鐃緒申鐃重�鐃緒申分鐃緒申鐃緒申 $k^n$ 鐃緒申与鐃緒申鐃緒申譴随申箸鐃緒申鐃緒申鐃緒申里鐃緒申箸鐃銃縁申佑任鐃緒申襦�
1.
$V$ 鐃熟器申鐃緒申任鐃緒申襦�
2.
$I(V)$ 鐃緒申 $R=k[X_1,\dots,X_n]$ 鐃緒申鐃叔ワ申鐃叔ワ申鐃緒申任鐃緒申襦�
3.
$R$ 鐃緒申鐃緒申弔慮鐃� $f_1,f_2$ 鐃叔￥申

\begin{displaymath}f_1,f_2 \notin I(V) \text{ かつ } f_1f_2\in I(V) \end{displaymath}

鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申里鐃渋醐申澆鐃緒申覆鐃緒申鐃�

鐃緒申鐃緒申 6.1   鐃緒申鐃塾器申 $S$ 鐃緒申 $0$ 鐃叔なわ申鐃緒申 $f_1,f_2$ 鐃叔￥申 $f_1f_2=0$ 鐃夙なわ申鐃塾わ申 鐃緒申鐃淑わ申鐃緒申鐃緒申 (鐃緒申鐃緒申鐃緒申 $X,Y,Z$ 鐃緒申多鐃準式鐃叔緒申表鐃緒申鐃緒申鐃叔わ申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申磴わ申鐃塾わ申 鐃緒申鐃緒申茲�鐃緒申鐃緒申鐃熟わ申鐃暑こ鐃夙￥申 (鐃銃￥申鐃塾醐申 $f_1,f_2$ 鐃祝つわ申鐃銃￥申鐃緒申鐃緒申鐃緒申 $f_1f_2=0$ 鐃緒申鐃緒申鐃緒申鐃緒申鐃緒申鐃夙わ申 鐃緒申鐃緒申鐃銃わ申鐃緒申鐃緒申鐃夙￥申 )

\begin{displaymath}S=\mbox{${\Bbb R}$}[X,Y,Z]/ \langle (X^2-Y^2-Z^2,Y-(\frac{1}{2}+X^2-Y^2-Z^2)X\rangle_{\text{ideal}} \end{displaymath}


Yoshifumi Tsuchimoto
2001-05-25