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¤U¤@­¶: Prime ideals ¤W¤@­¶: ¯S®íªº Ideals «e¤@­¶: ¯S®íªº Ideals

Principle ideals

¦b group ¤¤§Ú­Ì¤¶²Ð¹L cyclic subgroup, ¥¦¥i¥H¬O»¡¥]§t¬Y¤@­Ó¤¸¯Àªº³Ì¤pªº subgroup. ¦b ring ¤¤§Ú­Ì¤]¦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, §Ú­Ì­º¥ý·|¦Ò¼¥]§t¬Y¤@¤¸¯À¤§³Ì¤pªº ideal ¬°¦ó, ¦]¬°³o¬O³Ì²³æªº ideal. ­Yµ¹©w a $ \in$ 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 $ \in$ R ³o­Ó°²³]). ¥Ñ©ó 1 + 1 $ \in$ R, §Ú­Ì¥i¥H»¡¦s¦b¬Y¤@¤¸¯À $ \alpha$ $ \in$ R ¨Ï±o 2a = $ \alpha$ . a. ¦P²z¹ï¨ä¥Lªº¥¿¾ã¼Æ n, ¥Ñ©ó

na = $\displaystyle \underbrace{(1+\cdots+1)}_{n}^{}\,$ . a

©Ò¥H (ÁÂÁ 1 $ \in$ R ³o­Ó°²³]) ¦s¦b $ \beta$ $ \in$ R º¡¨¬ na = $ \beta$ . a. ¥t¤@¤è­±¥Ñ Lemma 6.1.2, ª¾ I ¤¤¤]¥²¶·¥]§t¹ï¥ô·Nªº r $ \in$ R, r . a ©M a . r ³oºØ¤¸¯À. µM¦Ó r . a = a . r (ÁÂÁ 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 $ \in$ R. ¥O A = {r . a | r $ \in$ R}, «h A ¬O R ªº¤@­Ó ideal. ¨Æ¹ê¤W, A ¬O R ¤¤¥]§t a ¤§³Ì¤pªº ideal.

µý ©ú. ±q«e­±ªº°Q½×§Ú­Ì¤wª¾: ­Y I ¬O R ¤¤¥]§t a ¤§³Ì¤pªº ideal, «h A $ \subseteq$ I. ¦]¦¹­Y¯àÃÒ±o A ¬O R ªº ideal, «hª¾ I = A.

§Ú­Ì§Q¥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' $ \in$ R. ¥Ñ©ó r . a - r' . a = (r - r') . a ¥B r - r' $ \in$ R, ª¾ r . a - r' . a $ \in$ A. ¥t¥ ¥ô¨ú R ¤¤¤@¤¸¯À r ¤Î A ¤¤¤@¤¸¯À r' . a, ¨ä¤¤ r' $ \in$ R. ¥Ñ©ó (r' . a) . r = r . (r' . a) = (r . r') . a ¥B r . r' $ \in$ R, ª¾ (r' . a) . r = r . (r' . a) $ \in$ A. ¦]¦¹ A ¬O R ªº ideal. $ \qedsymbol$

³q±`§Ú­Ì·|±N Lemma 6.5.1 ¤¤ªº A ¥Î $ \bigl($a$ \bigr)$ ¨Óªí¥Ü. ª`·N§Ú­Ì¬O¥Î¤j¤@ÂIªº¬A¸¹ $ \bigl($ $ \bigr)$ ¥H§K©M¤@¯ë¹Bºâ¶¡ªº¤p¬A¸¹ ( ) ²V²c.

Definition 6.5.2   °²³] R ¬O¤@­Ó commutative ring with 1, ¥B a $ \in$ R. «h

$\displaystyle \bigl($a$\displaystyle \bigr)$ = {r . a | r $\displaystyle \in$ R}

ºÙ¬° the principle ideal generated by a in R. ­Y I ¬° R ªº¤@­Ó ideal ¥B¦b R ¤¤¦s¦b¤@¤¸¯À a º¡¨¬ I = $ \bigl($a$ \bigr)$ «hºÙ I ¬O R ªº¤@­Ó principle ideal.

Example 6.5.3   ¦b $ \mathbb {Z}$ ¤¤, ¥ô¨ú n $ \in$ $ \mathbb {Z}$, «h©Ò¦³ n ªº­¿¼Æ©Ò¦¨ªº¶°¦X¬O¤@­Ó principle ideal, §Y $ \bigl($n$ \bigr)$ = {z . n | z $ \in$ $ \mathbb {Z}$}.

±N¨Ó§Ú­Ì·|¬Ý¨ì¦b $ \mathbb {Z}$ ¤¤©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§Ú­Ì¦³ $ \bigl($n$ \bigr)$ = $ \bigl($ - n$ \bigr)$), ¨Æ¹ê¤W§Ú­Ì¦³¥H¤Uªºµ²ªG.

Lemma 6.5.4   °²³] R ¬O¤@­Ó commutative ring with 1. ¦pªG ab $ \in$ R ¥B ¦s¦b¤@ unit u $ \in$ R º¡¨¬ a = u . b, «h $ \bigl($a$ \bigr)$ = $ \bigl($b$ \bigr)$.

µý ©ú. ¥Ñ©ó a = u . b, ¥Ñ©w¸qª¾ a $ \in$ $ \bigl($b$ \bigr)$. ¤S¥Ñ©ó $ \bigl($b$ \bigr)$ ¬O¤@­Ó ideal ¥B $ \bigl($a$ \bigr)$ ¬O¥]§t a ³Ì¤pªº ideal, ¬G±o $ \bigl($a$ \bigr)$ $ \subseteq$ $ \bigl($b$ \bigr)$. ¤Ï¤§, ¦] u ¬O R ªº unit, ¬G¦s¦b v $ \in$ R º¡¨¬ v . u = 1. ©Ò¥H¥Ñ b = (v . u) . b = v . a ª¾ b $ \in$ $ \bigl($a$ \bigr)$. ¦A§Q¥Î $ \bigl($b$ \bigr)$ ¬O¥]§t b ³Ì¤pªº ideal ±o $ \bigl($b$ \bigr)$ $ \subseteq$ $ \bigl($a$ \bigr)$. ¬GÃÒ±o $ \bigl($a$ \bigr)$ = $ \bigl($b$ \bigr)$. $ \qedsymbol$

¥H¤U¤¶²Ð¤@­Ó principle ideal ªºÂ²³æÀ³¥Î. §Ú­Ì¦b 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§Ú­Ì§PÂ_ 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. §Ú­Ì¤wª¾·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 $ \in$ R ¥B a$ \ne$ 0. §Ú­Ì¦Ò¼ $ \bigl($a$ \bigr)$ ³o¤@­Ó principle ideal. ¦]¬° a$ \ne$ 0 ¥B a $ \in$ $ \bigl($a$ \bigr)$, ¬Gª¾ $ \bigl($a$ \bigr)$$ \ne${0}. ¤£¹L¨Ì°²³] R ¤¤°£¤F {0} ©M R ¤w¥ ¨S¦³¨ä¥Lªº ideal, ¦]¦¹±o $ \bigl($a$ \bigr)$ = R. µM¦Ó 1 $ \in$ R, §Y 1 $ \in$ $ \bigl($a$ \bigr)$ ¬G¥Ñ $ \bigl($a$ \bigr)$ ªº©w¸qª¾¦s¦b r $ \in$ R ¨Ï±o 1 = r . a. ¤]´N¬O»¡ a ¬O¤@­Ó unit. $ \qedsymbol$

³Ì«á§Ú­Ì­n±j½Õ, ¦b Proposition 3.1.3 ¤¤§Ú­Ìª¾¹D¤@­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' $ \subseteq$ I. ¦pªG¤wª¾ I ¬O principle ideal, ³o¨Ã¤£«OÃÒ I' ·|¬O principle ideal.


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