¤U¤@¶: The Hamilton quaternions
¤W¤@¶: ¤@¨Ç Noncommutative Ring
«e¤@¶: ¤@¨Ç Noncommutative Ring
¥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
³oºØ§Î¦¡¨ä¤¤
a, b, c, d R. ¦]¬° R ¬O¤@Ó ring
§ÚÌ¥i¥H©w M2(R) ¤¤ªº¥[ªk©M¼ªk´N¬O¤@¯ë¯x°ªº¥[ªk©M¼ªk, §Y:
¦]¬° R ¬O¤@Ó ring with 1,
¤£Ãøµo²¥H¤Wªº¥[ªk©M¼ªk¨Ï±o
M2(R) ¦¨¬°¤@Ó ring, ¦Ó¥B
©M
¤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:
(ª`·N: ·í§ÚÌn»¡©ú¤@Ó ring R ¬O commutative ®É,
§ÚÌ¥²¶·ÃÒ©ú¹ï¥ô·Nªº a, b R ¬Ò¦³
a . b = b . a.
¤£¹LYn»¡©ú R ¬O noncommutative ®É, ¥un§ä¨ì¤@²Õ a, b R ¨Ï±o
a . bb . a §Y¥i.)
±q¤W±ªº¦¡¤l§Ú̪¾¹D
©M
¬O M2(R) ªº zero-divisor.
¦P®É¦b³oÓ¨Ò¤l¸Ì§Ṳ́]µo²¦b¤@Ó noncommutative ring ¤¤¬O¦³¥i¯àµo¥Í
a . b = 0 ¦ý
b . a 0 ªº²¶H.
±µ¤U¨Ó§ÚÌ·Q§ä¨ì M2(R) ¤¤©Ò¦³ªº zero-divisor ©M unit.
º¥ýÆ[¹î¥H¤Uªº¦¡¤l:
nª`·N§ÚÌ»Ýn R ¬O commutative ¦¡¤l (5.2) ¤ ·|¹ï.
¤j®aÀ³¸Ó¹ï
a . d - b . c ³oÓȤ£¯¥Í, ¥¦¬O
ªº determinant. ³q±`µ¹¤@¯x°
A M2(R) §ÚÌ¥Î
det(A) ªí¥Ü¨ä determinant. ¥Ñ©ó R ¬O¤@Ó ring, ©Ò¥H¹ï¥ô·Nªº
A M2(R), §Ú̳£¥i±o
det(A) R. Determinant
ÁÙ¦³¥H¤U³oÓ«nªº©Ê½è:
det(A . B) = det(A) . det(B), A, B 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 = ¦ý det(A) = 0 ®É, ¥Ñ©ó
, ¥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 = ¥B
det(A) = ¬O R ªº¤@Ó zero-divisor. ³]
0 ¬O R ¤¤¤@¤¸¯Àº¡¨¬
. = 0.
¦³¥H¤U¨âºØ¥i¯àµo¥Í:
(1)
a . ,
b . ,
c . ©M
d . ³£µ¥©ó 0: ¦¹®É¥O
B = , ¦p¦¹¤@¨Ó¦]
0, ©Ò¥H
B, ¦ý¬O
¦]¦¹¦b³oÓ±¡§Î®É, A ¬O M2(R) ªº¤@Ó zero-divisor.
(2)
a . ,
b . ,
c . ©M
d .
¤£¥þ¬° 0:
«h§Ú̦Ҽ
µM¦Ó
¥Ñ¦¡¤l (5.2) ª¾
©Ò¥H¦b³oÓ±¡ªp A ÁÙ¬O M2(R) ªº¤@Ó zero-divisor.
¨º»ò·í
A = ¥B
det(A) = ¤£¬O R ªº zero-divisor ®É¤S·|«ç¼Ë©O? °²³]¦s¦b
B = (¤]´N¬O»¡ a', b', c' ©M d' ¤£¥þ¬° 0) º¡¨¬
A . B = . ¦¹®É¦Ò¼
C = , «h¥Ñ¦¡¤l (5.2) ª¾
¦]¬° ¤£¬O zero-divisor ¥B a', b', c' ©M
d' ¤£¥þ¬° 0, ©Ò¥Hª¾
. a',
. b',
. c' ©M
. d' ¤£¥þ¬° 0. ¤]´N¬O»¡
(C . A) . B. ³o©M
(
C . A)
. B =
C . (
A . B) =
¬Û¥Ù¬Þ, ©Ò¥H¤£¥i¯à§ä¨ì
B º¡¨¬
A . B = . ¦P²z¥iª¾¤£¥i¯à§ä¨ì
B º¡¨¬
B . A = . ©Ò¥H A µ´¹ï¤£·|¬O M2(R) ªº¤@Ó zero-divisor.
¦]¦¹§Ú̱oÃÒ:
Proposition 5.5.1
Y
R ¬O¤@Ó commutative ring ¥B
A 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 §Ú̪¾¦b
M2() ©M
M2() ¤¤
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?
§Ú̦³¥H¤Uªºµ²ªG:
Proposition 5.5.2
Y
R ¬O¤@Ó
commutative ring with 1 ¥B
A 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 M2(
R) º¡¨¬
§Q¥Î¦¡¤l (
5.3) ±o
det(A) . det(B) = det(B) . det(A) = 1.
µM¦Ó
det(
A), det(
B)
R, ¬G±o det(
A) ¬O
R ªº¤@Ó unit.
¤Ï¤§, Y
A = ¥B
det(A) = ¬O R ªº¤@Ó unit, «h¦Ò¼
¦]
R, §Ú̪¾
B M2(
R). §Q¥Î¦¡¤l
(
5.2), ¥i±o
¦]¦¹
A ¬O
M2(
R) ªº¤@Ó unit.
¥Ñ Proposition 5.5.2 ª¾¦b
M2() ¤¤±©¦³ determinant ¬O
±1 ªº¯x°¤ ·|¬O unit, ¦Ó¦b
M2() ¤¤©Ò¦³ determinant ¤£¬O
0 ªº¯x°³£·|¬O unit.
¤U¤@¶: The Hamilton quaternions
¤W¤@¶: ¤@¨Ç Noncommutative Ring
«e¤@¶: ¤@¨Ç Noncommutative Ring
Administrator
2005-06-18