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¤U¤@­¶: The Hamilton quaternions ¤W¤@­¶: ¤@¨Ç Noncommutative Ring «e¤@­¶: ¤@¨Ç Noncommutative Ring

Matrix ring M2(R)

¥O R ¬O¤@­Ó commutative ring with 1. ¦Ò¼¶°¦X M2(R) ¬O©Ò¦³«Y¼Æ¦b R ªº 2×2 ¯x°©Ò¦¨ªº¶°¦X, ¤]´N¬O»¡ M2(R) ¤¤ªº¤¸¯À³£¬O

$\displaystyle \left(\vphantom{\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$\displaystyle \left.\vphantom{\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$

³oºØ§Î¦¡¨ä¤¤ a, b, c, d $ \in$ R. ¦]¬° R ¬O¤@­Ó ring §Ú­Ì¥i¥H©w M2(R) ¤¤ªº¥[ªk©M­¼ªk´N¬O¤@¯ë¯x°ªº¥[ªk©M­¼ªk, §Y:

$\displaystyle \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ + $\displaystyle \left(\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
a+a' & b+b' \\
c+c' & d+d' \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a+a' & b+b' \\
c+c' & d+d' \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a+a' & b+b' \\
c+c' & d+d' \\
\end{array}%%
}\right)$    ©M

$\displaystyle \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
a\cdot a'+b\cdot c' & a\cdot ...
... d' \\
c\cdot a'+d\cdot c' & c\cdot b'+d\cdot d' \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a\cdot a'+b\cdot c' & a\cdot b'+b\cdot d' \\
c\cdot a'+d\cdot c' & c\cdot b'+d\cdot d' \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a\cdot a'+b\cdot c' & a\cdot ...
... d' \\
c\cdot a'+d\cdot c' & c\cdot b'+d\cdot d' \\
\end{array}%%
}\right)$.

¦]¬° R ¬O¤@­Ó ring with 1, ¤£Ãøµo²¥H¤Wªº¥[ªk©M­¼ªk¨Ï±o M2(R) ¦¨¬°¤@­Ó ring, ¦Ó¥B $ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ ©M $ \left(\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%%
}\right)$ ¤À§O¬O M2(R) ªº 0 ©M 1. ©Ò¥H»¡ M2(R) ¬O¤@­Ó ring with 1. ¤£¹L§Y¨Ï R ¬O commutative, M2(R) ¤]¤£·|¬O commutative ring. ³o¥i¥Ñ¥H¤Uªº¨Ò¤l¬Ý¥X:

$\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right)$    ¦ý¬O    $\displaystyle \left(\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$.

(ª`·N: ·í§Ú­Ì­n»¡©ú¤@­Ó ring R ¬O commutative ®É, §Ú­Ì¥²¶·ÃÒ©ú¹ï¥ô·Nªº a, b $ \in$ R ¬Ò¦³ a . b = b . a. ¤£¹L­Y­n»¡©ú R ¬O noncommutative ®É, ¥u­n§ä¨ì¤@²Õ a, b $ \in$ R ¨Ï±o a . b$ \ne$b . a §Y¥i.)

±q¤W­±ªº¦¡¤l§Ú­Ìª¾¹D $ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
1 & 0 \\
\end{array}%%
}\right)$ ©M $ \left(\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ ¬O M2(R) ªº zero-divisor. ¦P®É¦b³o­Ó¨Ò¤l¸Ì§Ú­Ì¤]µo²¦b¤@­Ó noncommutative ring ¤¤¬O¦³¥i¯àµo¥Í a . b = 0 ¦ý b . a$ \ne$ 0 ªº²¶H.

±µ¤U¨Ó§Ú­Ì·Q§ä¨ì M2(R) ¤¤©Ò¦³ªº zero-divisor ©M unit. ­º¥ýÆ[¹î¥H¤Uªº¦¡¤l:

$\displaystyle \left(\vphantom{ \begin{array}{cc} a & b \\  c & d \\  \end{array}<tex2html_comment_mark>86 }\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$\displaystyle \left.\vphantom{ \begin{array}{cc} a & b \\  c & d \\  \end{array}<tex2html_comment_mark>86 }\right)$ . $\displaystyle \left(\vphantom{ \begin{array}{cc} d & -b \\  -c & a \\  \end{array}<tex2html_comment_mark>88 }\right.$$\displaystyle \begin{array}{cc} d & -b \\  -c & a \\  \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{cc} d & -b \\  -c & a \\  \end{array}<tex2html_comment_mark>88 }\right)$ = $\displaystyle \left(\vphantom{ \begin{array}{cc} a\cdot d-b\cdot c & 0 \\  0 & a\cdot d-b\cdot c \\  \end{array}<tex2html_comment_mark>90 }\right.$$\displaystyle \begin{array}{cc} a\cdot d-b\cdot c & 0 \\  0 & a\cdot d-b\cdot c \\  \end{array}$$\displaystyle \left.\vphantom{ \begin{array}{cc} a\cdot d-b\cdot c & 0 \\  0 & a\cdot d-b\cdot c \\  \end{array}<tex2html_comment_mark>90 }\right)$. (5.2)

­nª`·N§Ú­Ì»Ý­n R ¬O commutative ¦¡¤l (5.2) ¤ ·|¹ï. ¤j®aÀ³¸Ó¹ï a . d - b . c ³o­Ó­È¤£­¯¥Í, ¥¦¬O $ \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$ \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ ªº determinant. ³q±`µ¹¤@¯x° A $ \in$ M2(R) §Ú­Ì¥Î det(A) ªí¥Ü¨ä determinant. ¥Ñ©ó R ¬O¤@­Ó ring, ©Ò¥H¹ï¥ô·Nªº A $ \in$ M2(R), §Ú­Ì³£¥i±o det(A) $ \in$ R. Determinant ÁÙ¦³¥H¤U³o­Ó­«­nªº©Ê½è:

det(A . B) = det(A) . det(B),    $\displaystyle \forall$ A, B $\displaystyle \in$ M2(R). (5.3)

¨ì©³ M2(R) ¤¤¦³­þ¨Ç zero-divisor ©O? ¦P¾Ç¥i¯à·Q¨ì determinant ¬° 0 ªº¤¸¯À. ¨S¿ù, ·í A = $ \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$ \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ ¦ý det(A) = 0 ®É, ¥Ñ©ó $ \left(\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right.$$ \begin{array}{cc}
d & -b \\
-c & a \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right)$$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$, ¥Ñ¦¡¤l (5.2) ª¾ A ¬O¤@­Ó zero-divisor.

ÁÙ¦³¨S¦³¨ä¥Lªº zero-divisor ©O? ¨ä¹ê·í det(A) ¬O R ªº zero-divisor ®É, A ¤]·|¬O M2(R) ªº zero-divisor. ³o¬O¦]¬°¦pªG A = $ \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$ \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ ¥B det(A) = $ \alpha$ ¬O R ªº¤@­Ó zero-divisor. ³] $ \beta$$ \ne$ 0 ¬O R ¤¤¤@¤¸¯Àº¡¨¬ $ \alpha$ . $ \beta$ = 0. ¦³¥H¤U¨âºØ¥i¯àµo¥Í:

(1) a . $ \beta$, b . $ \beta$, c . $ \beta$ ©M d . $ \beta$ ³£µ¥©ó 0: ¦¹®É¥O B = $ \left(\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right.$$ \begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right)$, ¦p¦¹¤@¨Ó¦] $ \beta$$ \ne$ 0, ©Ò¥H B$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$, ¦ý¬O

A . B = $\displaystyle \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
a\cdot\beta & b\cdot\beta \\
c\cdot\beta & d\cdot\beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a\cdot\beta & b\cdot\beta \\
c\cdot\beta & d\cdot\beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a\cdot\beta & b\cdot\beta \\
c\cdot\beta & d\cdot\beta \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$.

¦]¦¹¦b³o­Ó±¡§Î®É, A ¬O M2(R) ªº¤@­Ó zero-divisor.

(2) a . $ \beta$, b . $ \beta$, c . $ \beta$ ©M d . $ \beta$ ¤£¥þ¬° 0: «h§Ú­Ì¦Ò¼

B = $\displaystyle \left(\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc} d & -b \\  -c & a \\  \end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
d\cdot\beta & -b\cdot\beta \\
-c\cdot\beta & a\cdot\beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
d\cdot\beta & -b\cdot\beta \\
-c\cdot\beta & a\cdot\beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
d\cdot\beta & -b\cdot\beta \\
-c\cdot\beta & a\cdot\beta \\
\end{array}%%
}\right)$$\displaystyle \ne$$\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$.

µM¦Ó

A . B = $\displaystyle \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc} d & -b \\  -c & a \\  \end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right)$,

¥Ñ¦¡¤l (5.2) ª¾

A . B = $\displaystyle \left(\vphantom{
\begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\beta & 0 \\
0 & \beta \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
\alpha\cdot\beta & 0 \\
0 & \alpha\cdot\beta \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\alpha\cdot\beta & 0 \\
0 & \alpha\cdot\beta \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\alpha\cdot\beta & 0 \\
0 & \alpha\cdot\beta \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$.

©Ò¥H¦b³o­Ó±¡ªp A ÁÙ¬O M2(R) ªº¤@­Ó zero-divisor.

¨º»ò·í A = $ \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$ \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ ¥B det(A) = $ \alpha$ ¤£¬O R ªº zero-divisor ®É¤S·|«ç¼Ë©O? °²³]¦s¦b B = $ \left(\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right.$$ \begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right)$$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ (¤]´N¬O»¡ a', b', c' ©M d' ¤£¥þ¬° 0) º¡¨¬ A . B = $ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$. ¦¹®É¦Ò¼ C = $ \left(\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right.$$ \begin{array}{cc}
d & -b \\
-c & a \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
d & -b \\
-c & a \\
\end{array}%%
}\right)$, «h¥Ñ¦¡¤l (5.2) ª¾

(C . A) . B = $\displaystyle \left(\vphantom{
\begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\alpha & 0 \\
0 & \alpha \\
\end{array}%%
}\right)$ . $\displaystyle \left(\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
a' & b' \\
c' & d' \\
\end{array}%%
}\right)$ = $\displaystyle \left(\vphantom{
\begin{array}{cc}
\alpha\cdot a' & \alpha\cdot b' \\
\alpha\cdot c' & \alpha\cdot d' \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\alpha\cdot a' & \alpha\cdot b' \\
\alpha\cdot c' & \alpha\cdot d' \\
\end{array}$$\displaystyle \left.\vphantom{

\begin{array}{cc}
\alpha\cdot a' & \alpha\cdot b' \\
\alpha\cdot c' & \alpha\cdot d' \\
\end{array}%%
}\right)$.

¦]¬° $ \alpha$ ¤£¬O zero-divisor ¥B a', b', c' ©M d' ¤£¥þ¬° 0, ©Ò¥Hª¾ $ \alpha$ . a', $ \alpha$ . b', $ \alpha$ . c' ©M $ \alpha$ . d' ¤£¥þ¬° 0. ¤]´N¬O»¡ (C . A) . B$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$. ³o©M

(C . A) . B = C . (A . B) = $\displaystyle \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$

¬Û¥Ù¬Þ, ©Ò¥H¤£¥i¯à§ä¨ì B$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ º¡¨¬ A . B = $ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$. ¦P²z¥iª¾¤£¥i¯à§ä¨ì B$ \ne$$ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$ º¡¨¬ B . A = $ \left(\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right.$$ \begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
0 & 0 \\
0 & 0 \\
\end{array}%%
}\right)$. ©Ò¥H A µ´¹ï¤£·|¬O M2(R) ªº¤@­Ó zero-divisor. ¦]¦¹§Ú­Ì±oÃÒ:

Proposition 5.5.1   ­Y R ¬O¤@­Ó commutative ring ¥B A $ \in$ M2(R), «h A ¬O M2(R) ªº¤@­Ó zero-divisor ­Y¥B°ß­Y det(A) = 0 ©Î det(A) ¬O R ªº¤@­Ó zero-divisor.

¥Ñ Proposition 5.5.1 §Ú­Ìª¾¦b M2($ \mathbb {Z}$) ©M M2($ \mathbb {Q}$) ¤¤ determinant ¬° 0 ªº¯x°·|¬O zero-divisor, ¦Ó determinant ¤£¬° 0 ªº¯x°´N¤£·|¬O zero-divisor.

·í R ¬O commutative ring with 1 ®É M2(R) ·|¦³­þ¨Ç unit ©O? §Ú­Ì¦³¥H¤Uªºµ²ªG:

Proposition 5.5.2   ­Y R ¬O¤@­Ó commutative ring with 1 ¥B A $ \in$ M2(R), «h A ¬O M2(R) ªº¤@­Ó unit ­Y¥B°ß­Y det(A) ¬O R ªº¤@­Ó unit.

µý ©ú. °²³] A ¬O M2(R) ªº¤@­Ó unit, «h¦s¦b B $ \in$ M2(R) º¡¨¬

A . B = B . A = $\displaystyle \left(\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%%
}\right)$.

§Q¥Î¦¡¤l (5.3) ±o

det(A) . det(B) = det(B) . det(A) = 1.

µM¦Ó det(A), det(B) $ \in$ R, ¬G±o det(A) ¬O R ªº¤@­Ó unit.

¤Ï¤§, ­Y A = $ \left(\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right.$$ \begin{array}{cc}
a & b \\
c & d \\
\end{array}$$ \left.\vphantom{
\begin{array}{cc}
a & b \\
c & d \\
\end{array}%%
}\right)$ ¥B det(A) = $ \alpha$ ¬O R ªº¤@­Ó unit, «h¦Ò¼

B = $\displaystyle \left(\vphantom{
\begin{array}{cc}
\alpha^{-1}\cdot d & \alpha^{...
...-b) \\
\alpha^{-1}\cdot(-c) & \alpha^{-1}\cdot a \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
\alpha^{-1}\cdot d & \alpha^{-1}\cdot (-b) \\
\alpha^{-1}\cdot(-c) & \alpha^{-1}\cdot a \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
\alpha^{-1}\cdot d & \alpha^{...
...-b) \\
\alpha^{-1}\cdot(-c) & \alpha^{-1}\cdot a \\
\end{array}%%
}\right)$.

¦] $ \alpha^{-1}_{}$ $ \in$ R, §Ú­Ìª¾ B $ \in$ M2(R). §Q¥Î¦¡¤l (5.2), ¥i±o

A . B = B . A = $\displaystyle \left(\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%%
}\right.$$\displaystyle \begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}$$\displaystyle \left.\vphantom{
\begin{array}{cc}
1 & 0 \\
0 & 1 \\
\end{array}%%
}\right)$.

¦]¦¹ A ¬O M2(R) ªº¤@­Ó unit. $ \qedsymbol$

¥Ñ Proposition 5.5.2 ª¾¦b M2($ \mathbb {Z}$) ¤¤±©¦³ determinant ¬O ±1 ªº¯x°¤ ·|¬O unit, ¦Ó¦b M2($ \mathbb {Q}$) ¤¤©Ò¦³ determinant ¤£¬O 0 ªº¯x°³£·|¬O unit.


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