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¤U¤@­¶: ¯S®íªº Ideals ¤W¤@­¶: ¤¤¯Å Ring ªº©Ê½è «e¤@­¶: Ring Homomorphism

¤T­Ó Ring Isomorphism ©w²z

©M group ¤@¼Ë, ring ¤]¦³¤T­Ó isomorphism ©w²z. ¥Ñ©ó§Ú­Ì¦³²¦¨ªº group isomorphism ©w²z¥i¥Î, ³o¤T­Ó isomorphism ©w²z´X¥G¥i¥Hª½±µ±À±o, §Ú­Ì¥u­nÅçÃÒ­¼ªk³¡¤À§Y¥i.

Definition 6.4.1   ¦pªG¨â­Ó rings R ©M R' ¶¡§A¥i¥H§ä¨ì¤@­Ó ring homomorphism ¬O isomorphism (§Y 1-1 ¥B onto), «h§Ú­ÌºÙ R ©M R' ³o¨â­Ó ring ¬O isomorphic, °O¬°: R $ \simeq$ R'.

Theorem 6.4.2 (First Isomorphism Theorem)   ­Y $ \phi$ : R$ \to$R' ¬O¤@­Ó ring homomorphism, «h

R/ker($\displaystyle \phi$) $\displaystyle \simeq$ im($\displaystyle \phi$).

µý ©ú. ­º¥ýª`·N¥Ñ Lemma 6.3.3 ª¾ im($ \phi$) ¬O¤@­Ó ring ¥B ker($ \phi$) ¬O R ªº ideal, ©Ò¥H R/ker($ \phi$) ¤]¬O¤@­Ó ring. §Q¥Î©M²Ä¤@­Ó group isomorphism ©w²z¬Û¦Pªº¤èªk, §Ú­Ì¦b R/ker($ \phi$) ³o¤@­Ó quotient ring ©M im($ \phi$) ³o­Ó ring ¤§¶¡§ä¨ì¤@­Ó¨ç¼Æ. ¦A»¡©ú³o­Ó¨ç¼Æ¬O ring homomorphism, ³Ì«á¦AÅçÃÒ¥¦¬O 1-1 ¥B onto.

§Ú­Ì¥i¥H§Q¥Î $ \phi$ »s³y¥H¤Uªº¨ç¼Æ:

$\displaystyle \psi$ : R/ker($\displaystyle \phi$)$\displaystyle \to$im($\displaystyle \phi$);    $\displaystyle \overline{a}$ $\displaystyle \mapsto$ $\displaystyle \phi$(a),  $\displaystyle \forall$ $\displaystyle \overline{a}$ $\displaystyle \in$ R/ker($\displaystyle \phi$).

§Ú­Ì­º¥ý»¡©ú $ \psi$ ¬O¤@­Ó`¦n¨ç¼Æ' (well defined function): ¦pªG a, b $ \in$ R ¨Ï±o $ \overline{a}$ ©M $ \overline{b}$ ¦b R/ker($ \phi$) ¤¤¬O¬Û¦Pªº. §Ú­Ì¥²¶·»¡©ú $ \phi$(a) = $ \phi$(b). ÁöµM a$ \ne$b, ¤£¹L¥Ñ $ \overline{a}$ = $ \overline{b}$ ª¾ a ©M b ¦b¥H ker($ \phi$) ³o­Ó ideal ªº¤ÀÃþ¤U¬O¦PÃþªº. §O§Ñ¤F a ©M b ¦PÃþªí¥Ü a - b $ \in$ ker($ \phi$). ¤]´N¬O»¡ $ \phi$(a - b) = 0. ¦A§Q¥Î $ \phi$ ¬O ring homomorphism ªº°²³], §Ú­Ì±o $ \phi$(a) - $ \phi$(b) = $ \phi$(a - b) = 0. §Y $ \phi$(a) = $ \phi$(b). ©Ò¥H§Ú­Ì»s³yªº $ \psi$ ¬O¤@­Ó well defined function.

±µ¤U¨ÓÃÒ $ \psi$ ¬O¤@­Ó ring homomorphism: ¹ï¥ô·Nªº $ \overline{a}$, $ \overline{b}$ $ \in$ R/ker($ \phi$), §Ú­Ì¦³

$\displaystyle \psi$($\displaystyle \overline{a}$ + $\displaystyle \overline{b}$) = $\displaystyle \psi$($\displaystyle \overline{a+b}$) = $\displaystyle \phi$(a + b)    ¥B    $\displaystyle \psi$($\displaystyle \overline{a}$ . $\displaystyle \overline{b}$) = $\displaystyle \psi$($\displaystyle \overline{a\cdot b}$) = $\displaystyle \phi$(a . b).

¥t¤@¤è­±¦]¬° $ \phi$ ¬O ring homomorphism, ©Ò¥H

$\displaystyle \phi$(a + b) = $\displaystyle \phi$(a) + $\displaystyle \phi$(b) = $\displaystyle \psi$($\displaystyle \overline{a}$) + $\displaystyle \psi$($\displaystyle \overline{b}$)    ¥B    $\displaystyle \phi$(a . b) = $\displaystyle \phi$(a) . $\displaystyle \phi$(b) = $\displaystyle \psi$($\displaystyle \overline{a}$) . $\displaystyle \psi$($\displaystyle \overline{b}$).

µ²¦X¥H¤W¤G¦¡, §Ú­Ì¥i±o

$\displaystyle \psi$($\displaystyle \overline{a}$ + $\displaystyle \overline{b}$) = $\displaystyle \psi$($\displaystyle \overline{a}$) + $\displaystyle \psi$($\displaystyle \overline{b}$)    ¥B    $\displaystyle \psi$($\displaystyle \overline{a}$ . $\displaystyle \overline{b}$) = $\displaystyle \psi$($\displaystyle \overline{a}$) . $\displaystyle \psi$($\displaystyle \overline{b}$).

§Ú­Ì³Ì«á­nÃÒ©ú $ \psi$ ¬O 1-1 ¥B onto. ³o¨ä¹ê¤£¥²ÃÒ¤F(·íµM§A­n¦h¦¹¤@Á|¤]¨SÃö«Y), ¦]¬°§Ú­Ì¦b Theorem 2.6.1 ¤wÃÒ¹L $ \psi$ ³o­Ó¨ç¼Æ¦b¥[ªk¬Ý¦¨¬O group homomorphism ¤w¸g¬O 1-1 ¥B onto.

Á`µ²: §Ú­ÌÃÒ±o¤F $ \psi$ ¬O¤@­Ó±q G/ker($ \phi$) ¨ì im($ \phi$) ªº isomorphism. ©Ò¥H G/ker($ \phi$) $ \simeq$ im($ \phi$). $ \qedsymbol$

·íµM¤F¦pªG©w²z¤¤ªº $ \phi$ ¬O onto. ¨º»ò§Ú­Ìª¾ im($ \phi$) = R'. ¦]¦¹§Ú­Ì¦³¥H¤Uªº¤Þ²z:

Corollary 6.4.3   ­Y $ \phi$ : R$ \to$R' ¬O¤@­Ó onto ªº ring homomorphism, «h

R/ker($\displaystyle \phi$) $\displaystyle \simeq$ R'.

²¦b§Ú­Ì¨Ó¬Ý¬Ý ring ªº²Ä¤G­Ó isomorphism ©w²z. ¥¦À³¸Ó¬O«ç¼Ëªº§Î¦¡©O? §Ú­Ì¥ý¦^ÅU¤@¤U group ªº±¡ªp: µ¹©w¤@ group G, ­Y H ¬O G ªº subgroup ¥B N ¬O G ªº normal subgroup. «h H $ \cap$ N ¬O H ªº normal subgroup, ¥B H/(H $ \cap$ N) $ \simeq$ (H . N)/N. ¦n²¦b§Ú­Ì§â group ´«¦¨ ring, subgroup ´«¦¨ subring, normal subgroup ´«¦¨ ideal, ³Ì«á§O§Ñ¤F±N­¼§ï¬°¥[.

Theorem 6.4.4 (Second Isomorphism Theorem)   ­Y R ¬O¤@­Ó ring, S ¬O R ªº subring ¥B I ¬O R ªº ideal, «h S $ \cap$ I ¬O S ªº ideal, ¥B

S/(S $\displaystyle \cap$ I) $\displaystyle \simeq$ (S + I)/I.

µý ©ú. ­º¥ýª`·Nªº¬O¥Ñ Lemma 6.2.1 ª¾ S + I ¬O R ªº subring, ¥B I $ \subseteq$ S + I ¦]¦¹ª¾ I ¬O S + I ªº ideal (½Ð°Ñ¦Ò 6.2 ¸`ªº³Ì«á). ©Ò¥H (S + I)/I ½T¹ê¬O¤@­Ó ring.

¦p¦P¦b group ªº±¡ªp, §Ú­Ì·Q¥Î first isomorphism ©w²z¨ÓÃÒ©ú¦¹©w²z. §Ú­Ì¥ý§ä¤@­Ó±q S ¨ì (S + I)/I ªº¨ç¼Æ. ¦Ò¼ $ \phi$ : S$ \to$(S + I)/I, ¨ä¤¤¹ï©Ò¦³ªº s $ \in$ S §Ú­Ì¦³ $ \phi$(s) = $ \overline{s}$.

²¦b­nÃÒ $ \phi$ ¬O¤@­Ó ring homomorphism. ¨Æ¹ê¤W¹ï¥ô·Nªº s, s' $ \in$ S, §Ú­Ì¦³

$\displaystyle \phi$(s + s') = $\displaystyle \overline{s+
s'}$ = $\displaystyle \overline{s}$ + $\displaystyle \overline{s'}$ = $\displaystyle \phi$(s) + $\displaystyle \phi$(s')    ¥B    $\displaystyle \phi$(s . s') = $\displaystyle \overline{s\cdot
s'}$ = $\displaystyle \overline{s}$ . $\displaystyle \overline{s'}$ = $\displaystyle \phi$(s) . $\displaystyle \phi$(s').

§Q¥Î Theorem 2.6.4 ªºÃÒ©ú, §Ú­Ì±o $ \phi$ : S$ \to$(S + I)/I ¬O onto. ¦]¦¹¥i¥H¥Î First Isomorphism Theorem (Corollary 6.4.3) ±o¨ì

S/ker($\displaystyle \phi$) $\displaystyle \simeq$ (S + I)/I.

¬Æ»ò¬O ker($ \phi$) ©O? ¨Ì©w¸q ker($ \phi$) ¬O S ¤¤ªº¤¸¯À s ¨Ï±o $ \phi$(s) ¬O (S + I)/I ªº identity, $ \overline{0}$. ¤]´N¬O»¡ $ \phi$(s) = $ \overline{s}$ = $ \overline{0}$. §O§Ñ¤F $ \overline{s}$ = $ \overline{0}$ ªí¥Ü s - 0 = s $ \in$ I. ¥Ñ¦¹ª¾ ker($ \phi$) ªº¤¸¯À¬J­n¦b S ¤¤¤]­n¦b I ¤¤; ´«¥y¸Ü»¡ ker($ \phi$) $ \subseteq$ S $ \cap$ I. ¤Ï¤§­Y a $ \in$ S $ \cap$ I, «h¦] a $ \in$ I ±o $ \phi$(a) = $ \overline{a}$ = $ \overline{0}$. ¬G S $ \cap$ I $ \subseteq$ ker($ \phi$). ¥Ñ¦¹ª¾ ker($ \phi$) = S $ \cap$ I. ¦]¦¹§Ú­Ì¥Ñ Lemma 6.3.3 ª¾ S $ \cap$ I ¬O S ªº ideal ¤]¥Ñ First Isomorphism Theorem ª¾

S/(S $\displaystyle \cap$ I) $\displaystyle \simeq$ (S + I)/I.

$ \qedsymbol$

³Ì«á§Ú­Ì¨Ó¬Ý²Ä¤T­Ó isomorphism ©w²z. ¦P¼Ëªº, ±N Theorem 2.6.5 ¤¤ªº group ´«¦¨ ring ¤Î normal subgroup ´«¦¨ ideal, §Ú­Ì¦³¥H¤U¤§²Ä¤T isomorphism ©w²z:

Theorem 6.4.5 (Third Isomorphism Theorem)   ­Y $ \phi$ : R$ \to$R' ¬O¤@­Ó onto ªº ring homomorphism. °²³] J' ¬O R' ªº¤@­Ó ideal. ¥O

J = {a $\displaystyle \in$ R | $\displaystyle \phi$(a) $\displaystyle \in$ J'}.

«h J ¬O R ªº ideal ¥B

R/J $\displaystyle \simeq$ R'/J'.

µý ©ú. §Ú­Ì©w $ \psi$ : R$ \to$R'/J', º¡¨¬ $ \psi$(a) = $ \overline{\phi(a)}$$ \forall$ a $ \in$ R.

¥Ñ $ \phi$ ¬O ring homomorphism ª¾

$\displaystyle \psi$(a + b) = $\displaystyle \overline{\phi(a+
b)}$ = $\displaystyle \overline{\phi(a)+\phi(b)}$ = $\displaystyle \overline{\phi(a)}$ + $\displaystyle \overline{\phi(b)}$ = $\displaystyle \psi$(a) + $\displaystyle \psi$(b)

¥B

$\displaystyle \psi$(a . b) = $\displaystyle \overline{\phi(a\cdot
b)}$ = $\displaystyle \overline{\phi(a)\cdot\phi(b)}$ = $\displaystyle \overline{\phi(a)}$ . $\displaystyle \overline{\phi(b)}$ = $\displaystyle \psi$(a) . $\displaystyle \psi$(b).

¬G $ \psi$ ¬O¤@­Ó±q R ¨ì R'/J' ªº ring homomorphism.

¦p«e, §Ú­Ì¥i¥Î Theorem 2.6.5 ªºÃÒ©úª¾ $ \psi$ : R$ \to$R'/J' ¬O¤@­Ó onto ªº ring homomorphism, §Ú­Ì¦A¦¸¥Î First Isomorphism Theorem ª¾

R/ker($\displaystyle \psi$) $\displaystyle \simeq$ R'/J'.

¬Æ»ò¬O ker($ \psi$) ©O? ­Y a $ \in$ ker($ \psi$) §Y $ \psi$(a) = $ \overline{\phi(a)}$ = $ \overline{0}$, ¤]´N¬O»¡ $ \phi$(a) ©M 0 ¦b¥Î J' ªº¤ÀÃþ¤U¬O¦PÃþªº. ©Ò¥H $ \phi$(a) - 0 = $ \phi$(a) $ \in$ J'. ¥Ñ J ªº©w¸qª¾, ³oªí¥Ü a $ \in$ J. ¬G ker($ \psi$) $ \subseteq$ J. ¥t¥ ­Y a $ \in$ J, «h $ \phi$(a) $ \in$ J' ¬G¦b R'/J' ¤¤ $ \psi$(a) = $ \overline{\phi(a)}$ = $ \overline{0}$. ¦]¦¹ a $ \in$ ker($ \psi$), ±o J $ \subseteq$ ker($ \psi$). ¤]´N¬O»¡ ker($ \psi$) = J ¥B¥Ñ Lemma 6.3.3 ª¾ J ¬O R ªº ideal (¨ä¹ê§Ú­Ì¦b Theorem 6.3.5 ¤wª¾ J ¬O R ªº ideal). $ \qedsymbol$

³Ì«á§Ú­Ì§Q¥Î Correspondence Theorem ¨Ó¬Ý Third Isomorphism Theorem ªº¤@­Ó¯S®íª¬ªp. ¥O I ¬O R ªº ideal, $ \phi$ : R$ \to$R/I ¬O©w¸q¦¨ $ \phi$(a) = $ \overline{a}$ ³o­Ó onto ªº ring homomorphism. ¥ô·N R/I ¤¤ªº ideal J' ¥Ñ«e Corollary 6.3.7 ª¾¬O¥Ñ R ¤¤ªº¬Y¤@ ideal J §Q¥Î $ \phi$ ±o¨ì: ¤]´N¬O»¡ J' = $ \phi$(J) = J/I. ¬G¥Ñ Theorem 6.4.5 §Ú­Ì¦³¥H¤Uªº©w²z(¦³ªº®Ñ¬OºÙ³o­Ó¬° Third Isomorphism Theorem.)

Theorem 6.4.6 (Third Isomorphism Theorem)   ­Y R ¬O¤@­Ó ring, I ¬O R ªº¤@­Ó ideal. «h R/I ¤¤ªº¥ô¤@ ideal ³£¬O J/I ³oºØ§Î¦¡, ¨ä¤¤ I $ \subseteq$ J ¥B J ¬O R ªº ideal. ¦Ó¥B§Ú­Ì¦³

(R/I)/(J/I) $\displaystyle \simeq$ R/J.

µý ©ú. ¥ô¤@ R/I ªº ideal ³£¬O J/I ³oºØ§Î¦¡¤w¦b Corollary 6.3.7 ÃÒ±o. ¦Ó

(R/I)/(J/I) $\displaystyle \simeq$ R/J

¥i¥Ñ Theorem 6.4.5 ª½±µ±o¨ì. ¤]´N¬O¥N: R' = R/I, J' = J/I ¥B¦Ò¼ $ \phi$ : R$ \to$R/I, ²Å¦X $ \phi$(a) = $ \overline{a}$. ¦¹®É¥i±o J = {a $ \in$ R | $ \phi$(a) $ \in$ J'}. ¬G¥Ñ R/J $ \simeq$ R'/J' ±oÃÒ. $ \qedsymbol$


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¤U¤@­¶: ¯S®íªº Ideals ¤W¤@­¶: ¤¤¯Å Ring ªº©Ê½è «e¤@­¶: Ring Homomorphism
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