¤U¤@¶: ¯S®íªº Ideals
¤W¤@¶: ¤¤¯Å Ring ªº©Ê½è
«e¤@¶: Ring Homomorphism
©M group ¤@¼Ë, ring ¤]¦³¤TÓ isomorphism ©w²z. ¥Ñ©ó§Ú̦³²¦¨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§Ú̺Ù
R ©M
R' ³o¨âÓ ring ¬O
isomorphic, °O¬°:
R R'.
Theorem 6.4.2 (First Isomorphism Theorem)
Y
:
RR' ¬O¤@Ó ring homomorphism, «h
µý ©ú.
º¥ýª`·N¥Ñ Lemma
6.3.3 ª¾
im(
) ¬O¤@Ó ring ¥B
ker(
) ¬O
R ªº ideal, ©Ò¥H
R/ker(
) ¤]¬O¤@Ó ring. §Q¥Î©M²Ä¤@Ó group isomorphism
©w²z¬Û¦Pªº¤èªk, §Ú̦b
R/ker(
) ³o¤@Ó quotient ring ©M
im(
) 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 »s³y¥H¤Uªº¨ç¼Æ:
:
R/ker(
)
im(
);
(
a),
R/ker(
).
§Ú̺¥ý»¡©ú
¬O¤@Ó`¦n¨ç¼Æ' (well defined
function): ¦pªG
a,
b R ¨Ï±o
©M
¦b
R/ker(
) ¤¤¬O¬Û¦Pªº. §ÚÌ¥²¶·»¡©ú
(
a) =
(
b). ÁöµM
ab, ¤£¹L¥Ñ
=
ª¾
a ©M
b ¦b¥H
ker(
) ³oÓ ideal ªº¤ÀÃþ¤U¬O¦PÃþªº. §O§Ñ¤F
a ©M
b ¦PÃþªí¥Ü
a -
b ker(
). ¤]´N¬O»¡
(
a -
b) = 0. ¦A§Q¥Î
¬O ring
homomorphism ªº°²³], §Ú̱o
(
a) -
(
b) =
(
a -
b) = 0. §Y
(
a) =
(
b). ©Ò¥H§ÚÌ»s³yªº
¬O¤@Ó well defined
function.
±µ¤U¨ÓÃÒ ¬O¤@Ó ring homomorphism: ¹ï¥ô·Nªº
,
R/ker(), §Ú̦³
(
+
) =
(
) =
(
a +
b) ¥B
(
. ) =
(
) =
(
a . b).
¥t¤@¤è±¦]¬°
¬O ring homomorphism, ©Ò¥H
(
a +
b) =
(
a) +
(
b) =
(
) +
(
) ¥B
(
a . b) =
(
a)
. (
b) =
(
)
. (
).
µ²¦X¥H¤W¤G¦¡, §ÚÌ¥i±o
(
+
) =
(
) +
(
) ¥B
(
. ) =
(
)
. (
).
§Ú̳̫ánÃÒ©ú ¬O 1-1 ¥B onto.
³o¨ä¹ê¤£¥²ÃÒ¤F(·íµM§An¦h¦¹¤@Á|¤]¨SÃö«Y), ¦]¬°§Ú̦b Theorem
2.6.1 ¤wÃÒ¹L ³oÓ¨ç¼Æ¦b¥[ªk¬Ý¦¨¬O group homomorphism
¤w¸g¬O 1-1 ¥B onto.
Á`µ²: §ÚÌÃÒ±o¤F ¬O¤@Ó±q
G/ker() ¨ì
im() ªº
isomorphism. ©Ò¥H
G/ker() im().
·íµM¤F¦pªG©w²z¤¤ªº ¬O onto. ¨º»ò§Ú̪¾
im() = R'.
¦]¦¹§Ú̦³¥H¤Uªº¤Þ²z:
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 N ¬O H ªº
normal subgroup, ¥B
H/(H N) (H . N)/N. ¦n²¦b§Ú̧â
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 I ¬O
S ªº ideal, ¥B
S/(
S I)
(
S +
I)/
I.
µý ©ú.
º¥ýª`·Nªº¬O¥Ñ Lemma
6.2.1 ª¾
S +
I ¬O
R ªº subring, ¥B
I 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
: S(S + I)/I,
¨ä¤¤¹ï©Ò¦³ªº s S §Ú̦³
(s) = .
²¦bnÃÒ ¬O¤@Ó ring homomorphism. ¨Æ¹ê¤W¹ï¥ô·Nªº s, s' S, §Ú̦³
(
s +
s') =
=
+
=
(
s) +
(
s') ¥B
(
s . s') =
=
. =
(
s)
. (
s').
§Q¥Î Theorem 2.6.4 ªºÃÒ©ú, §Ú̱o
: S(S + I)/I ¬O onto.
¦]¦¹¥i¥H¥Î First Isomorphism Theorem (Corollary 6.4.3) ±o¨ì
S/ker(
)
(
S +
I)/
I.
¬Æ»ò¬O
ker(
) ©O? ¨Ì©w¸q
ker(
) ¬O
S ¤¤ªº¤¸¯À
s ¨Ï±o
(
s) ¬O (
S +
I)/
I ªº
identity,
. ¤]´N¬O»¡
(
s) =
=
. §O§Ñ¤F
=
ªí¥Ü
s - 0 =
s I. ¥Ñ¦¹ª¾
ker(
)
ªº¤¸¯À¬Jn¦b
S ¤¤¤]n¦b
I ¤¤; ´«¥y¸Ü»¡
ker(
)
S I. ¤Ï¤§Y
a S I, «h¦]
a I ±o
(
a) =
=
. ¬G
S I ker(
). ¥Ñ¦¹ª¾
ker(
) =
S I. ¦]¦¹§ÚÌ¥Ñ Lemma
6.3.3 ª¾
S I ¬O
S ªº ideal ¤]¥Ñ First Isomorphism
Theorem ª¾
S/(
S I)
(
S +
I)/
I.
³Ì«á§Ų́ӬݲĤTÓ isomorphism ©w²z. ¦P¼Ëªº, ±N Theorem 2.6.5
¤¤ªº group ´«¦¨ ring ¤Î normal subgroup ´«¦¨ ideal, §Ú̦³¥H¤U¤§²Ä¤T
isomorphism ©w²z:
Theorem 6.4.5 (Third Isomorphism Theorem)
Y
:
RR' ¬O¤@Ó onto ªº ring homomorphism. °²³]
J' ¬O
R' ªº¤@Ó ideal. ¥O
«h
J ¬O
R ªº ideal ¥B
R/
J R'/
J'.
µý ©ú.
§ÚÌ©w
:
RR'/
J', º¡¨¬
(
a) =
,
a R.
¥Ñ ¬O ring homomorphism ª¾
(
a +
b) =
=
=
+
=
(
a) +
(
b)
¥B
(
a . b) =
=
=
. =
(
a)
. (
b).
¬G
¬O¤@Ó±q
R ¨ì
R'/
J' ªº ring homomorphism.
¦p«e, §ÚÌ¥i¥Î Theorem 2.6.5 ªºÃÒ©úª¾
: RR'/J' ¬O¤@Ó
onto ªº ring homomorphism, §Ú̦A¦¸¥Î First Isomorphism Theorem ª¾
R/ker(
)
R'/
J'.
¬Æ»ò¬O
ker(
) ©O? Y
a ker(
) §Y
(
a) =
=
, ¤]´N¬O»¡
(
a) ©M 0 ¦b¥Î
J' ªº¤ÀÃþ¤U¬O¦PÃþªº. ©Ò¥H
(
a) - 0 =
(
a)
J'. ¥Ñ
J ªº©w¸qª¾, ³oªí¥Ü
a J. ¬G
ker(
)
J. ¥t¥ Y
a J, «h
(
a)
J' ¬G¦b
R'/
J' ¤¤
(
a) =
=
. ¦]¦¹
a ker(
), ±o
J ker(
). ¤]´N¬O»¡
ker(
) =
J
¥B¥Ñ Lemma
6.3.3 ª¾
J ¬O
R ªº ideal (¨ä¹ê§Ú̦b Theorem
6.3.5 ¤wª¾
J ¬O
R ªº ideal).
³Ì«á§Ú̧Q¥Î Correspondence Theorem ¨Ó¬Ý Third Isomorphism Theorem
ªº¤@Ó¯S®íª¬ªp. ¥O I ¬O R ªº ideal,
: RR/I ¬O©w¸q¦¨
(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 ±o¨ì: ¤]´N¬O»¡
J' = (J) = J/I. ¬G¥Ñ Theorem
6.4.5 §Ú̦³¥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 J ¥B
J ¬O
R ªº ideal.
¦Ó¥B§Ú̦³
(
R/
I)/(
J/
I)
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)
R/
J
¥i¥Ñ Theorem
6.4.5 ª½±µ±o¨ì. ¤]´N¬O¥N:
R' =
R/
I,
J' =
J/
I ¥B¦Ò¼
:
RR/
I, ²Å¦X
(
a) =
. ¦¹®É¥i±o
J = {
a R |
(
a)
J'}.
¬G¥Ñ
R/
J R'/
J' ±oÃÒ.
¤U¤@¶: ¯S®íªº Ideals
¤W¤@¶: ¤¤¯Å Ring ªº©Ê½è
«e¤@¶: Ring Homomorphism
Administrator
2005-06-18