¤U¤@¶: Prime ideals
¤W¤@¶: ¯S®íªº Ideals
«e¤@¶: ¯S®íªº Ideals
¦b group ¤¤§Ṳ́¶²Ð¹L cyclic subgroup,
¥¦¥i¥H¬O»¡¥]§t¬Y¤@Ó¤¸¯Àªº³Ì¤pªº subgroup. ¦b ring ¤¤§Ṳ́]¦³©Ò¿×ªº
principle ideal, ¥¦¬O¥]§t¬Y¤@¤¸¯Àªº³Ì¤pªº ideal.
°²³] R ¬O¤@Ó commutative ring with 1. n¤F¸Ñ R ¤¤ªº ideal
ªø¬Æ»ò¼Ë¤l, §Ú̺¥ý·|¦Ò¼¥]§t¬Y¤@¤¸¯À¤§³Ì¤pªº ideal ¬°¦ó,
¦]¬°³o¬O³Ì²³æªº ideal. Yµ¹©w a R, «h¥]§t a ªº³Ì¤p ideal I
À³¸Óªø¬Æ»ò¼Ë¤l©O? º¥ý I ¦Ü¤Ön¥]§t a ©Ò²£¥Íªº¥[ªkªº cyclic
group, §Y
{0, a, - a, 2a, - 2a,..., na, - na,...}. ª`·N«e±´£¹L³o¸Ì
2a ¤£¬O 2 . a ¦Ó¬O
(1 + 1) . a (§O§Ñ¤F 1 R ³oÓ°²³]).
¥Ñ©ó 1 + 1 R, §ÚÌ¥i¥H»¡¦s¦b¬Y¤@¤¸¯À
R ¨Ï±o
2a = . a. ¦P²z¹ï¨ä¥Lªº¥¿¾ã¼Æ n, ¥Ñ©ó
na =
. a
©Ò¥H (ÁÂÁÂ 1 R
³oÓ°²³]) ¦s¦b
R º¡¨¬
na = . a. ¥t¤@¤è±¥Ñ Lemma
6.1.2, ª¾ I ¤¤¤]¥²¶·¥]§t¹ï¥ô·Nªº r R, r . a ©M
a . r ³oºØ¤¸¯À. µM¦Ó
r . a = a . r (ÁÂÁÂ R ¬O
commutative ring ³oÓ°²³]), ¦]¦¹ I ¤¤¦Ü¤Ön¥]§t©Ò¦³ªº r . a
³oºØ§Î¦¡ªº¤¸¯À. ¦pªG¥Ñ©Ò¦³ªº r . a ³o¼Ëªº¤¸¯À©Ò¦¨ªº¶°¦X¬O R
ªº¤@Ó ideal, ¨º»ò¥¦¦ÛµM´N¬O¥]§t a ªº³Ì¤p ideal ¤F.
Lemma 6.5.1
°²³]
R ¬O¤@Ó commutative ring with 1, ¥B
a R. ¥O
A = {
r . a |
r R}, «h
A ¬O
R ªº¤@Ó ideal. ¨Æ¹ê¤W,
A
¬O
R ¤¤¥]§t
a ¤§³Ì¤pªº ideal.
µý ©ú.
±q«e±ªº°Q½×§Ṳ́wª¾: Y
I ¬O
R ¤¤¥]§t
a ¤§³Ì¤pªº ideal, «h
A I. ¦]¦¹Y¯àÃÒ±o
A ¬O
R ªº ideal, «hª¾
I =
A.
§Ú̧Q¥Î Lemma 6.1.2 ¨ÓÃÒ©ú A ¬O R ªº ideal. ¥ô¨ú A
¤¤¨â¤¸¯À r . a ©M r' . a, ¨ä¤¤ r, r' R. ¥Ñ©ó
r . a - r' . a = (r - r') . a ¥B r - r' R, ª¾
r . a - r' . a A. ¥t¥ ¥ô¨ú R ¤¤¤@¤¸¯À r ¤Î A ¤¤¤@¤¸¯À r' . a, ¨ä¤¤
r' R. ¥Ñ©ó
(r' . a) . r = r . (r' . a) = (r . r') . a ¥B
r . r' R, ª¾
(r' . a) . r = r . (r' . a) A. ¦]¦¹ A ¬O R ªº ideal.
³q±`§ÚÌ·|±N Lemma 6.5.1 ¤¤ªº A ¥Î
a ¨Óªí¥Ü.
ª`·N§Ú̬O¥Î¤j¤@ÂIªº¬A¸¹
¥H§K©M¤@¯ë¹Bºâ¶¡ªº¤p¬A¸¹
( ) ²V²c.
Definition 6.5.2
°²³]
R ¬O¤@Ó commutative ring with 1, ¥B
a R. «h
ºÙ¬° the
principle ideal
generated by
a in
R. Y
I ¬°
R ªº¤@Ó ideal ¥B¦b
R
¤¤¦s¦b¤@¤¸¯À
a º¡¨¬
I =
a «hºÙ
I ¬O
R ªº¤@Ó
principle ideal.
Example 6.5.3
¦b
¤¤, ¥ô¨ú
n , «h©Ò¦³
n ªº¿¼Æ©Ò¦¨ªº¶°¦X¬O¤@Ó
principle ideal, §Y
n = {
z . n |
z }.
±N¨Ó§ÚÌ·|¬Ý¨ì¦b
¤¤©Ò¦³ªº ideal ³£¬O principle ideal,
¤£¹L³o¹ï¤@¯ëªº ring ¨Ã¤£¤@©w¹ï. ¥t¥ Y I ¬O¤@Ó principle ideal,
¨Ã¤£ªí¥Ü²£¥Í I ªº¤¸¯À¬O°ß¤@ªº (¨Ò¦p«e±ªº¨Ò¤l§Ú̦³
n = - n), ¨Æ¹ê¤W§Ú̦³¥H¤Uªºµ²ªG.
Lemma 6.5.4
°²³]
R ¬O¤@Ó commutative ring with 1. ¦pªG
a,
b R ¥B
¦s¦b¤@ unit
u R º¡¨¬
a =
u . b, «h
a =
b.
µý ©ú.
¥Ñ©ó
a =
u . b, ¥Ñ©w¸qª¾
a b. ¤S¥Ñ©ó
b ¬O¤@Ó ideal ¥B
a ¬O¥]§t
a ³Ì¤pªº
ideal, ¬G±o
a b. ¤Ï¤§, ¦]
u ¬O
R ªº unit, ¬G¦s¦b
v R º¡¨¬
v . u = 1. ©Ò¥H¥Ñ
b = (
v . u)
. b =
v . a ª¾
b a. ¦A§Q¥Î
b
¬O¥]§t
b ³Ì¤pªº ideal ±o
b a.
¬GÃÒ±o
a =
b.
¥H¤U¤¶²Ð¤@Ó principle ideal ªºÂ²³æÀ³¥Î. §Ú̦b lemma 6.2.4
¤¤ª¾¹D: ·í R ¬O¤@Ó division ring ®É, R ¤¤¥u¦³ {0} ©M R
³o¨âÓ ideals. ·í R ¬O¤@Ó field ®É (R ¤]´N¬O¤@Ó division
ring), R ·íµM¤]´N¨S¦³ nontrivial proper ideal. ·í R ¬O
commutative ring with 1 ®É, ³o¬O¤@ÓÀ°§U§Ú̧PÂ_ R ¬O§_¬°¤@Ó
field ªº¦n¤èªk.
Proposition 6.5.5
Y
R ¬O¤@Ó commutative ring with 1, «h
R ¬O¤@Ó field
Y¥B°ßY
R ¨S¦³ nontrivial proper ideal.
µý ©ú.
§Ṳ́wª¾·í
R ¬O¤@Ó field ®É,
R ¨S¦³ nontrivial proper ideal.
¤Ï¤§, ¦pªG
R ¨S¦³ nontrivial proper ideal, §ÚÌ·QÃÒ©ú
R ¬O¤@Ó
field. ¥Ñ©ó
R ¤w°²³]¬O commutative ring with 1,
¨Ì©w¸q§ÚÌ¥unÃÒ©ú
R ¤¤«D 0 ªº¤¸¯À³£¬O unit. ¥ô¨ú
a R ¥B
a 0. §Ú̦Ҽ
a ³o¤@Ó principle ideal. ¦]¬°
a 0 ¥B
a a, ¬Gª¾
a{0}.
¤£¹L¨Ì°²³]
R ¤¤°£¤F {0} ©M
R ¤w¥ ¨S¦³¨ä¥Lªº ideal, ¦]¦¹±o
a =
R. µM¦Ó 1
R, §Y
1
a ¬G¥Ñ
a ªº©w¸qª¾¦s¦b
r R ¨Ï±o
1 =
r . a. ¤]´N¬O»¡
a ¬O¤@Ó unit.
³Ì«á§ÚÌn±j½Õ, ¦b Proposition 3.1.3 ¤¤§Ú̪¾¹D¤@Ó cyclic
group ¤¤ªº subgroup ³£¬O cyclic group. ¤£¹L¹ï principle ideal,
³o´N¤£¤@©w¹ï¤F. ¤]´N¬O»¡Y I, I' ³£¬O R ªº ideal ¥B
I' I. ¦pªG¤wª¾ I ¬O principle ideal, ³o¨Ã¤£«OÃÒ I'
·|¬O principle ideal.
¤U¤@¶: Prime ideals
¤W¤@¶: ¯S®íªº Ideals
«e¤@¶: ¯S®íªº Ideals
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2005-06-18