Z E S Z Y T Y N A U K O W E W Y Ż S Z E J S Z K O Ł Y P E D A G O G IC Z N E J W B Y D G O S Z C Z Y P r o b le m y M a te m a ty c zn e 1982 z.4 A n d r z e j N o w ic k i U M K T o urń R y s z a r d Ż u c h o w s k i W S P B y d g o s z c z S P E C T R A L S P A C E S A N D R A D IC A L S IN S Y S T E M S O P ID E A L S 1. In tro d u ction . In th is p a p e r , w e s tu d y s e t s o f id e a ls o f a com m utative rin g R with id en tity, w h ic h a r e c lo s e d u n d or a n y In t e r s e - c tio n s a n d c o n ta in in g R . If E s a t is fie s th e s e c o n d itio n s , then b y E w e d e n o te the s e t o f a ll E -p rim e id e a ls l . e j d e a l s P in E , fo r w h ic h the co n d itio n A B С P , w h e r e A , В b e lo n g to E , im p lies A c . P o r В С P . W e In tro d u c e in E a t o p o lo g y a n d w e p r o v e , that i f E s a t is fie s an a d d itio n a l co n d itio n , then th e re e x is t s a rin g S , s u c h that the s p a c e s S p e c S a n d E a r e h om eom orph ie.
S im u lta n eo u sly, w e g e t s o m e p r o p e r tie s s e - c a lle d r£ - r a d ic a l id e a ls i.e . id e a ls o f E , w h ich a r e in te r s e c tio n s o f E -p rim e id a ls . F u rth er, w e ex a m in e s y s te m s o f id e a ls In the s e n s e [ 8 ] , [9] , and w e g iv e a n a p p lic a tio n o f th is t h e o r y fo r a d e s c r ip tio n o f so m e
d istin g u ish s u b s e ts o f th e s e s y s te m s a n d fo r a d e s c r ip tio n o f r a d ic a ls , w h ic h a r e c o n n e c t e d with th e s e s u b s e ts .
T h e s e s u b s e ts : A ( R , M ) , B ( R , M ) , C ( R , M ) w e r e in tro d u c e d in [ 8] , [9 ] . W e s h a ll p r o v e , that if w e g iv e an a d d itio n a l assu m ption , e v e r y o f th e s e c la s s is a s p e c t r a l s p a c e .
T h ro u g h o u t th is p a p e r, a ll r in g s a r e com m utative w ith unity. L e t R b e a r in g . B y i( R ) w e d e n o te the s e t o f a ll id e a ls in R . If T is a n y s u b s e t o f R , then b y r ( T ) w e d e n o te a r a d ic a l o f T i.e . an in te r s e c tio n o f a ll prim e id e a ls c o n ta in in g T . A n id e a l A o f R is r a d ic a l iff r ( A ) - A . If A Is an id e a l o f OO R , a n d x £ R , then b y A w e d e n o te the id e a l ( A t x n ) w h e re X n - o
A lB . I r g R i
Л
r * ь е А . It i s e a s y to v e r i f y that r ( A )=b Ê В J
» ( r ( A ) : x ) . b e t ( R , M ) b e a s y s te m o f id e a ls in the s e n s e 1 7 ] , H . [ » ] . I d e a l e o f M w ill b e ca U ed M -id e a ls .
A n M - id e a l P is prim itive, if th e re e x is t s a multi p lic a t iv e ly c lo s e d s u b s e t S o f R s u c h that P n S •» <p, and P is m axim al a m o n g
M -Id e a ls o f R , d is jo in t from S . T h e s e t o f prim itive M - id e a ls w ill b e d e n o te d b y B ( R , M )
A n M -id e a l P is c a lle d M -p rim e, If fo r M - id e a ls A , В the co n d itio n А .В С P , im p lies A С P o r В с:' P , T h e s e t o f a ll M -p rim e id e a ls o f ( R , M ) w ill b e d e n o te d b y C ( R , M ) . M o r e o v e r , b y a (R , m) w ill b e d e n o te d the s e t o f a ll prim e id e a ls o f R, w h ic h b e lo n g to M .
B y so m e m o d ificatio n s o f the p r o o f o f the th eorem at [5] ,
fo r d ifferen tia l rin g s , o n e c a n p r o v e , that A ( R , M )c H (R , m) С C (R , M j.
If T is a n y s u b s e t o f R , then b y ^t] w e d e n o te the s m a lle s t M -id e a l co n ta in in g T . If A is a n id e a l o f R , then b y A w e d e n o te the
w g r e a te s t M -id e a l co n ta in ed in A .
2. E -p rim e id e a ls . b et R b e a rin g, l ( R ) - the s e t o f a ll id e a ls o f R , and le t £ b e a s u b s e t o f l ( R ) s u c h that R £ E . I d e a ls o f E a r e c a lle d E - id e a ls . A n E - id e a l P f R is c a lle d E -p rim e, i f fo r E - id e a ls A , B , the co n d itio n A .B С P , im p lies A C P o r В С P .
*
T h e s e t o f a ll E -p rim e id e a ls w ill b e d e n o te d b y E . If T is a n y s u b s e t o f R , then the in te r s e c tio n o f a ll E -p rim e id e a ls , w h ich contain s T , w ill b e d e n o te d b y rE ( T ) , and w ill b e c a lle d an E - r a d ic a l
o f T . f l f th e re is n o E -p r im e id e a ls c o n ta in in g T , then w e s e t : rE ( T ) - R.
A n E - id e a l A w ill b e c a lle d E - r a d lc a l iff rE ( A ) - A .
P R O P O S IT IO N 2.1. If T , S a r e s u b s e ts o f R , A , В a r e E *4 d ea ls a n d P £ E * then ( 1 ) T С rE ( T ) ( 2 ) T С P if and o n ly if r£ ( T ) C . P ( 3 ) U T С S , then rE ( T ) c . rE ( s ) ( 4 ) ГЕ ( rE ^ ^ ^ " rE ^ T ^
( 5 ) rE ( P ) - P ( 6 ) i*E ( A ♦ В ) - r E ( r E ( A ) ♦ гЕ ( в ) ) ( 7 ) ге ( А В ) - ге ( А п В ) - г Е ( А ) Г. rE ( B ) PROOF4. . ( l ) , ( 2 ) , ( З ) , ( 5 ) a re ob v iou s. ( 4 ) . S in ce ( 2 ) , w e h ave [ p £ E * P О т ] - [ р £ E * Р Э rE ( T ) j , • ° ге ^ге ^т ^ “ [ p é е! p d г е ( т ) } “ / | { Р ^ Е* Р ^ 5 Т \ " гЕ ( Т ) .
( 6 ) . S in ce ( з ) , w e h ave the inclusion Ç_ . Now , if P Э A + В, then P D A and P D В , thus P 3 r E ( A ) and P D «*Е ( В ) . F in ally, we
get Р Э rE ( a ) , ♦ r £ ( B ) .
( 7 ) . B y ( 3 ) , w e get rE ( A B ) c rE ( A ^ B ) c гЕ (a )>*\ rE ( В ) , It su ffic e s to sh ow , that i-£ ( a b ) - ге ( а ) n гЕ ( в ) . Łet I - £ P £ E ^ P D A B I , J - | p £ ET PZ5 a | , К - I P f Е ^ Р Э в ] . S in ce I - J U K , w e h a ve r _ ( A B ) - 1 n P L- n p .
P 6 I P € J U К
= Г Л р - П О р - r (a ) n г ( в ) .
P e u р е к E E
If T is an y s u b s e t of R, then b y VE ( T ) w e sh a ll denote the s e t of all E -prim e id e a ls containing T . T h e condition ( 2 ) o f P roposition 2.1 implies VE ( T ) - VE ( ( T ) ) - VE ( r E ( T ) ) .
P R O P O S IT IO N 2.2. If E is a se t of id e a ls in R, R (= E and E is c lo s e d under an y intersections, then
( 1 ) VE ( 0 ) - E , VE ( l ) - ^ ( 2 ) If i f i I » * family of s u b s e t s of R then n v E ( T , ) * VE ( ^ T i 5 i £ i I « ; i ( 3 ) If T Jt T 2 a r e s u b s e ts o f R, then U VE ( T 2 ) “ “ V rE < T l> Ге ( Т 2 )> L e t E d I ( R ) b e a s e t c lo s e d u n d er a n y in te r s e c tio n s and
containing R. B y the a b o v e Proposition, В i s a topological s p a c e with the c lo s e d s e t s VE ( T ) . T h e op en s e t s ln E h av e the form
( T ) ■ ^ P e E ? P ^ > T ] , w h e re T Is a s u b s e t of R. If У С E? w e denote J£ (y ) - f l j p , P É y ] , If Y i* ^ and Je ( Y ) = - R, If Y - It is c le a r, that JE ( v ) £ Б . P R O P O S IT IO N 2.3. ( a ) If T с R. then Je Ve ( t ) . rE ( T ) it . . ( b ) If Y C E , then VeJe ( Y ) - Y , w h e re Y is the c lo s u re of Y ln E . P R O O F . ( a ) Je Ve ( T ) - n { P . p
e
VE ( T )J
- p i [ p €• E ,^ P 3 t } - rE ( T ) . ( b ) S in c e Y С Ve (Je (y ) ) , w e h a v e Y с VeJe (y) . C o n v e rse ly , If P G E * an d P О Je(y) , an d w e a ssu m e that P ^ Y , then there e x is ts a n d E -Id e a l A such, that D £ (a ) n Y - P G- DE (/0T h e n w e h a v e Y C v E ( A )t h en ce P 3 Je ( Y ) ;o A an d P £- VE (,A ) It contradicts with the fact that P £ De (a ) .
C O R O L L A R Y 2.4. If P G E ? then VE ( P ) - -[p] .
C O R O L L A R Y 2.5. T h e re is a bijection betw een the se t of
~*r
c lo s e d s u b s e ts of E an d the se t of E -r a d lc a l E -id e a ls in R. T h e m appings VE , J£ a re o r d e r -r e v e r s in g bijections.
P R O P O S IT IO N 2.6. И PC- E ^ t h e n V£ (p) i s a non -empty irred u cib le c lo s e d s u b s e t of E . E v e r y non-em pty Irred u cible c lo s e d s u b s e t o f E h a s the form VE ( P ) , w h ere P G еЛ
P R O O F . Let P É E r a n d let VE ( P ) С v E ( rE ( A ) ) О v E ( rE ( B ) ) -
T h e n P 3 rE ( A ) rE ( B ) . w h ence P 3 ге ( а ) o r P 3 rE ( B ) s o we h a v e V£ ( P ) c: v E ( rE ( A ) ) o r v E ( p )<C v E ( rE ( B ) ) - S u p p o s e now, that
A-V is a non-empty c lo s e d irre d u cib le s u b s e t in E , and A-V - A-Ve ( q ) , w h ere Q £ E an d te ( q ) - Q. W e show, that Q £ E Г If Q 3 A B , w h ere А , В 6 E , then V£ ( q ) C Ve ( A B ) - Ve ( a ) u Ve ( B ) . S in ce
^1»
V (q) is irred u cible, Ve (q) c Vg. (a ) o r Ve (q) q Ve ( B ) , H e n c e w e h ave Q ^ A o r Q ^ B ,
A set E is s a id to b e rE -N o eth erian , if E sa tisfie s the a sc e n d in g chain condition on E -r a d ic a l id e a ls .
T h e follow ing lemma la o b v io u s.
L E M M A 2.7. Let E be a n rE -N o e th e ria n aet. И T la a aubaet of R , then there e x is ts a finite a e t T Q , au ch that T Q С T an d rE ( T )
“ r E ^ T 0 ^ *
P R O P O S IT IO N 2.8. If E la a n r„ -N o e th e ria n , then e v e r y op en
*• Ł
aet in E la quasi-com pact.
P R O O F , ^ i r a t w e prove, that if x £ R, then D E ( x ) la a n open qu asi-com pact se t In E . B e c a u s e for T C R, w e h a v e D r ( T ) -
D ft)
t £ T E it su ffic e s to sh ow , that If D _ ( x ) - DE ( x . ) , w h e re x. £ R
1 fc I
for e a c h I £ I, then I>E ( x ) - D E ( x £ ) и . . . u DE (*| )
B y the Lemma 2.7., w e h av e 1 n DE ( X ) - i V l ° E ( x i> - DE (^ Xi S i £ I } > “ DE ( r E ({ Xl » i € I } ^ “ " D E ^ r E ^ X i . ’ X i 3 r ) ) " D E ^ x i » • • • . x i
}
) ” D E ^ X i . ^ U " * I n 1 IT 1 , U D _ ( x . ) • N o w , if D ( A ) is a n y o p e n s e t in В 0 then В 1 в n DE < A > - DE ( r E ( A ) ) - DE < rE ( £ l ...a r i > ~ ° Е * * « $ -= DE ( a 1 )u . . . U D E ( a n ) , w h e r e a ^ a 2...a n€ ' A *Dj (a' ,a s a fin ite u nion o f q u a s i-c o m p a c t o p e n s e t s , is q u a s i-c o m p a c t.
*
C O R O L L A R Y 2.9. If E is an r _ N o e th e ria n s e t , then E is В
a q u a s i-c o m p a c t s p a c e .
T H E O R E M 2.10. L e t E b e a s e t o f id e a ls o f R , c lo s e d u n d er a n y in te r s e c tio n s and R £ E . If E is an r _ - N o e th e ria n s e t , then th e re
E
e x is t s a rin g S su ch , that the s p a c e s S p e c g and E * a re h om eom orph ic.
P R O O F . A p p ly in g the re su lts of [ Ź ] , it su ffic e s to sh o w , that £ h a s the follow ing prop erties
a ) E * Is T Q - s p a c e . b ) E is qu asi-com pact,
. *r
c ) T h e q u asi-com p act op en s u b s e t s of E a re d o s e d u nder finite intersections.
\ *
d ; T h e qu a si-c o m p a c t op en s e t s in E form an op en b a s is . e ) E v e r y non-em pty ir r e d u d b le d o s e d s u b s e t h a s a g e n e ric
point.
T h e property a ) is o b v io u s an d i s satisfied b y a n y set E of id e a ls o f R, d o s e d u n d er intersections and w hich p o s s e s s e s R.
b ) - C o ro lla r y 2.9. c ) , d ) - P rop osition 2.8. e ) - P roposition 2.6.
N o w , w e g iv e som e applications o f the a b o v e theory for the s e ts : A ( R , M ) , B ( R , M ) , C ( R , M ) , w h ere (R , M ) i s a system of id e a ls in the s e n s L & ], [VJ .
3. T h e s p a c e a (R . M ) . bet (R , M ) b e a system o f id e a ls, an d let A ( R , M ) b e the se t of a ll prime id e a ls, w hich b e lo n g to M. M o re o v e r, let E • | G É M; r ( G ) » • T h e s e t E i s c lo s e d u nder a n y intersections an d R € E , s o E sa tisfie s the conditions o f the P a rt 2. W e sh all p ro v e , that E ^ - A ( R , M ) .
If T Is a n y s u b s e t o f R, then b y rA ( T ) w e denote the sm allest ra d ic a l M -id e a l in (R , M ) containing T .
L E M M A 3.1. К G 6 E , then ( GsT ) £• E , for e v e r y s u b s e t T of R.
P R O O F . Let x £ R. T h e n G x ( G : x ) an d r ( G s x ) f < G x ) -( r -( G ) : x )m -( G : x ) , s o -(G a c ) £ E . N o w , if T is a n y s u b s e t of R, then ( G : T ) ( G : x ) , and h en ce w e h ave ( G : T ) £ E .
X g T
P R O P O S IT IO N 3.2. If S , T a re s u b s e ts of R, then rA ( s ) n rA ( T ) * - rA ( S T ) .
P R O O F 4, (th is method is sim ila r to the method o f R.M . C oh n QQ ) . F ir s t, w e p r o v e that гд ( S ) r A ( T ) C гд ( S T ) . S in c e Т С ( г д ( 5 Т ) : S ) and ( г д ( S T ) : S ) e . E (L em m a 3 ,1 .), w e h a v e гд ( Т ) с ( г д ( S T ) : s ) , that m ean s S C ( r ( S T ) : r . ( T ) ) and h e n c e r . ( s ) C ( r . ( S T ) : r . ( T ) )
ie . rA ( Т ) г д ( S ) C r A ( S T ) F u rth er, w e h a v e ( r ( s ) O r ( T ) ) 2 - ( r ( s ) O r ( T ) ) ( r ( s ) O r (t ) ) A ' " ' A A ' 1 rA ( S ) 0 r A T h e in v e r s e in c lu s io n is o b v io u s . C r A ( S ) r A ( T ) C r A ( S T ) . that im pU es гд ( Б ) П гд ( т ) с г д ( S T ) T H E O R E M 3.3. If ( R . M ) is a s y s te m and E - -^ G C M ; r ( G ) - G ^ , then E * - A ( R , M ) . P R O O F . It s u ffc e s to p r o v e , that e v e r y id e a l in E is prim e, L e t P £ E * . а д е р , x y £ P . T h e n гд ( х ) г д ( у ) С г д ( х у ) С г д ( Р ) - P , and h e n c e гд ( x ) C P o r гд ( у ) С Р , that m ean s x £ P o r y £ P .
T H E O R E M 3.4. if (R .W l) is a s y s te m o f id e a ls and E - - [g € M , r ( G ) - g} then гд ( T ) - rE ( T ) , fo r e v e r y s u b s e t T o f R.
P R O O F . It s u ffic e s to v e r i f y ( b y T h e o re m 3 .3 .) that гд ( T ) - - П | р £ А ( R . M ) ; P D l j L e t G - гд ( Т ) . If G - R , then the th e s is is triv ia l. S u p p o s e n ow that G ^ R an d x R \ G . C o n s id e r an in d u c itiv e fa m ily 2 jx ” | h £ E ; G C H , x ^ H ^ J . L e t P b e a m axim al elem en t in ^ x * W e p r o v e , that P £ A ( R . M ) . T h e r e fo r e it s u ffic e s to d em on stra te, that P is a prim e id e a l in R. S u p p o s e , that u v £ P , ифр, v ^ P .
T h e n Р ^ г д ( P . u ) , ( P . v ) , s o
s < £ r A ( Р , и ) П г д ( P , v ) - гд ( ( Р , и ) ( P , v ) ) C r A ( P ) - P , it g iv e s a c o n tra d ic tio n with x ^ P . S o P i s a prim e id e a l, and w e p r o v e d that fo r e v e r y x £ R \ G th e re e x is ts a prim e id e a l P ^ Ê A f R . M ) s u c h that
x G C P X • F in a lly , w e h a v e G ■■ О *
B y the T h e o r e m s 2.10., 3.3 w e h a v e
C O R O L L A R Y 3.5. If ( R , M ) is a s y s te m w h ich s a t is fie s the a s c e n d in g ch a in co n d itio n on гд - r a d ic a l M -id e a ls , than th ere e x is t s *
a ring S au ch that a p a c a s A ( R , M ) an d S p « c S a re homeomorphlc. Other p r o o f» of C o ro lla ry 3.5, w e can find In the p a p e r [вД (f o r a p e c ia l ayatem a ) an d ["бД.
N o w , w e g iv e another description of the Id e a l гд ( т ) . If T la a n y a u baet of R, then we denote: { ^ [q “ T
W n + l - г < [ И n l ter n > O.
T H E O R E M 3 .« ^ Let (R , M ) b e a ayatem. If T la a a u b a e t of R, then rA ( T ) - U f r } n.
n—О
P R O O P . Let H - It la obvloua, that r ( H ) - H. If x e H. then x e ( т ] п d [ { T l nl . f®r aome n, therefore [x j С
с Г Ы „ ] с С' H. B y Induction, it la e a a y to prove, that { т 1 С
С гд ( ' 1') « tor e v e r y n <= N , h en ce H С rA ( T K
T H E O R E M 3.7. If (R , m ) la a ayatem, then fo r an y au baeta S , T of R, and for e v e r y n, m Ł N holda: { s } n f r l m C” ^ S T ^n+m
P R O O P . W e p ro v e thia theorem In s e v e r a l parte. a ) Pirat, w e pro v e that. If x e. R, then x [тД с г ( [ х Д ) .
S in c e T С ( [ х т ] : x ) <С ( [>тД : х П) £ М, во w e h a v e [тД С
1C*, п -о
с ! s Ł < M г хП) с r ( «so ^
s хП) ) ■ г ( [ х^
1
х )*
b ) W e demie na träte, that r ( s ) r ( £тД ) C r ( [s t | ) . B y the part a ) , w e h ave s j V J c г ( [ s i ] ) . H e n c e, It fo llo w » that r ( S ) r ( И ) с r ( s ) n л г ( [ т Д ) - r ( s [ T ] ) c r ( r ( L S T ] ) ) - r ( [ S T ] ) .
c ) B y induction, w e ah all pro ve , that ( s ) C.
j
S t | ^P o r n - O, the in c lu s io n is trivia l. S u p p o s e , that this in c lu s io n is s a tis fie d fo r so m e n. T h e n ( s ) \ T}n + l “ ( s ) r ( [ { T } n l > C
с r ( s ) r ( Ц т } п] ) С Г( [ s Щ ) С Г ( [ { з т } п] ) - { s T} n+1
d ) W e s e t o n e m. B y in du ction with r e s p e c t to n, w e p r o v e that { 4 A c { s 4 n . . n. F o r n - O, the In c lu s io n fo llo w s from c ) .
F-urth.r, w . h » v . { s l n ł l { т| m- г ( Г { з ' !п1 ) { т } т С r ( [ { s ] n M j ) C
С '• < Г Ы П, ПЛ ) - { s T } n łn itl
4. T h e s p a c e C ( R , M ) . Let ( R , M ) b e a system of Id e a ls and let C ( R t M ) b e the set of a ll M -prim e id e a ls in M . If T is a n y su b se t o f R, then b y r c ( T ) , we denote the intersection of all prime Id e a ls containing T . Let E ■» M . the se t E is c lo s e d u nder a n y intersections an d contains R, s o it sa tisfie s the assum ption of the part 2. U s in g e a rlie r notations, w e h a v e E » C ( R , M ) , rE ( T ) - r ^ ( T ) ,
B y the Th eorem 2.10. we h ave
C O R O L L A R Y 4.1. If (jR, m) is a system satisfyin g the a s c e n d in g chain condition on r _ -r a d i c a l M -id e a ls , then there e x ists a rin g S
%
s u c h that the s p a c e s C ( R , m) an d S p e c S a re homeomorphic. N o w , we give som e properties r ^ r a d ic a l id e a ls , that m eans s u c h M ld e a ls G , for w hich G r _ ( G ) . W e s a y , that an M id e a l is M irre d u
-/
c ib le , if it iant an intersection of two M -id e a ls , which p ro p erly contains Its.
P R O P O S IT IO N 4.2. E v e r y M -ir r e d u d b le r^ ,-rad ical Id e a l is M-prim e.
P R O O F . Let P b e an M -irre d u c lb le r^.—rad ic a l id e a l and let A , В be M -id e a ls su c h that A B C P . T h e n ( A + P ) ( В + P ) с P and
w e h a v e P - i"c(a + Р ) О г с ( в + P ) . In d e e d , P . rc ( P ) 3 rc Э г с ( А + P ) ( В + P ) ) - rc ( A + P ) О r c ( B + P ) o rc ( p ) . P. B e c a u s e P is M -irre d u c lb le , s o P m r ( a + P ) or p - r ( b + P ) .
c o
F in a lly A C P o r В О p .
P R O P O S IT IO N 4.3. If (R , M ) is a system satisfyin g the a s c e n ding chain condition on r ^ -r a d ic a l id e a ls, then e v e r y -r a d ic a l id e a l is a finite intersection of M -id e a ls .
P R O O F . S u p p o s e , that the se t 21 of a ll r ^ -r a d ic a l id e a ls which a re not finite intersections of M -id e a ls , i s non-empty. Let P be a
maximal element ot X • ВУ the P rop osition 4.2, P i s not M -irre d u cib ie. T h e re fo re , there e x ist M -id e a ls A , В su c h that P Ç A , P ç В an d A r> В ■ P. S in c e r ^ ( A ) , rc ( B ) d o n 't b e lon g to Z , then r^. ( a ) - Aj^ n А ^ л . . Aj^ rc ( B ) - В x г, В 2 r> . . . r>Bj w h e re A , . . . . A fc, B ^ . . . . Bj £ C ( R , M ) . N o w , w e h ave P - rc ( p ) “ rc ^ A n B ) “ rc ^ A ^ n rC ^ B ) " A l ° * * ° A krt r\ B 1 n . . . r\ Bj
this contradicts with the fact that P £ Z. .
C O R O L L A R Y 4.4. If (R , M ) is a system of id e a ls satisfyin g the a sc e n d in g chain condition on r ^ -r a d lc a l id e a ls then for e v e r y M -ld e a l A there e x ist only a finite s e t o f minimal M -prim e Id e a ls
w h ich c o n ta in A
5. T h e s p a c e B ( R , m) . Let ( R , M ) be a system of id e a ls and let B ( R , M ) b e the s e t of all primitive M -id e a ls .
P R O P O S IT IO N 5.1. T h e follow ing conditions a re equivalent ( 1 ) A £ B ( R , M )
( 2 ) A i s a prime id e a l a n d A - r ( A ) £-( 3 ) r ( A ) i s a prime id e a l an d A - r(A)^|_
( 4 ) T h e re e x is ts a prime id e a l P in R, s u c h that A
-P R O O -P . Is sim ilar to the proof
01
the a n a lo g o u s theorem for differential rin gs ( [4
] P ro p . 2 .2 ).If T is a n y s u b s e t of R, then b y rß ( т ) w e denote the intersection of all primitive M -id e a ls containing T.
T H E O R E M 5.2. If T i s a s u b s e t o f R, then rß ( T ) - г ( | т ] ) ^ P R O O P . If P i s a prime id e a l containing T , then by P rop osition 5.1. , P^j. is primitive and P ^ Z> И Z> T . T h e n r B ( T ) с г( И ) , an d w h en ce w e h ave r ß ( T ) С r ( [ t J ) ^ .
-If Q Is a primitive id e a l containing T , then by Proposition 5.2, w e h ave that r ( Q ) is a prime id e a l containing r ( [tT ) and - Q. T h e re fo re Q - t ( q ) #
3
г ( ( т ] ) г and finally r B ( T )3
r ( JtJ ) "•* s' ' C O R O L L A R Y 5.3. If G a n d H a re M -id e a ls , then r g ( G H ) — rB ( G H H ) - rB ( G ) O r B ( H ) . P R O O F . r0 ( G H ) - r ( [ g • h ] ) # - r ( G H ) ^ - ( g O H ) # - r(|GrnH] )^-- rB ( G n H ) . rB ( G H ) - г ( [ Ь н З ) ^ » r ( G H ) t t - ( r ( G ) n r (h) ) ^ -- r ( G ) # П r ( H ) # -- r ( [ G ] )# 0 r ( [ H ] ) # - r B ( G ) n rB ( H ) . A n M - id e a l G Is c a lle d r - r a d ic a l, if r (g ) - G . В вP R O P O S IT IO N 5.4. If ( R , M ) is a system s a tis fy in g the a s c e n d in g ch a in co n d itio n o n r - r a d ic a l id e a ls , then th e re e x is t s a rin g
В
S s u ch that the s p a c e s b ( R ,m ) and S p e c S h om eom orp h lc. T h e p r o o f is sim ila r to the a n a lo g o u s p r o o f o f the theorem fo r d iffe re n tia l rin gs .
R E F E R E N C E S
[1 ] R .M .C o h n , S y s te m s o f id e a ls , C a n a d . J. o f M ath. 2 l ( l 9 6 9 ) .
[2 ] M .H o c h s ter, P rim e id e a l stru ctu re in com m u tative r in g s , T r a n s ,
o f the A m e r ic a n M ath. S o c ie t y 1 4 2 (1 9 6 9 ) p. 43-60.
[3] W .F .K e ig h e r , On the q u a s i a ffin e s c h e m e o f a d iffe re n tia l rin g (t o a p p e a r ).
[4] W .F .K e ig h e r , Q u a si-p rim e id e a ls In d iffe re n tia l rin g s , H o u s to n J.
o f M ath. 4 (1 9 7 9 ) p. 379 -388.
[5] A .N o w ic k i, Q u a si-p rim e and d-prim e id e a ls in com m utative d iffe r e n tial r in g s (t o a p p e a r ).
[6] A .N o w ic k i, P rim e id e a ls s tru ctu re in , a d d itiv e c o n s e r v a t iv e s y s te m s (t o a p p e a r ).
[7] A .N o w ic k i, R .Ż u c h o w s k i, S o m e re m a rk s o n s y s te m s o f id e a ls (t o a p p e a r in P r o b le m y M a te m a ty c zn e , W S P B y d g o s z c z ) .
[8 ] A .N o w ic k i, R .Ż u c h o w s k i, S p e c ia l s y s te m s o f id e a ls in com m utative rin g s (t o a p p e a r in C om m en ta tion es M a th e m a tic a e ).
[9] R .Ż u c h o w s k i, S y s te m s o f id e a ls in com m utative rin g s , (t o a p p e a r in C om m en ta tio n es M a th e m a tic a e ).
P R Z E S T R Z E N I E S P E K T R A L N E O R A Z R A D Y K A Ł Y W S Y S T E M A C H ID E A Ł Ó W
S T R E S Z C Z E N IE
W n in ie js z e j p r a c y za jm u jem y s ię ro d zin a m i id e a łó w p ie r ś c ie n ia p r z e m ie n n e g o R z je d y n k ą , zam kn iętym i z e w z g lę d u n a d o w o ln e p r z e k r o je i z a w ie r a ją c y m i R. J e ż e li E je s t ta k ą r o d z in ą , to p r z e z E *
o z n a c z m y z b ió r w s z y s t k ic h id e a łó w E - p ie r w s z y c h . W p ro w a d z a m y w E * t o p o lo g ię i u dow adn iam y, ż e p r z y p ew n ym z a ło ż e n iu dodatkow ym is tn ie je p ie r ś c ie ń S taki, ż e p r z e s t r z e n ie S p e c S i E s ą h om eom orf- ic z n e .
J e d n o c z e ś n ie , o trzym u jem y s z e r e g w ła s n o ś c i tzw . id e a łó w rE - r a d y k a ln y c h , to z n a c z y ta k ich id e a łó w z E , k tó re s ą p r z e k ro ja m i id e a łó w E - p ie r w s z y c h . W d a ls z y m c ią g u k o n c en tru je m y s ię na s y s te m a c h id e a łó w w s e n s ie [ 8 ] , [ 9 ] i p o d a je m y z a s t o s o w a n ie p o w y ż s z e j te o rii d o o p is u p e w n y c h w y r ó ż n io n y c h p o d z b io r ó w ta k ich s y s te m ó w o r a z do o p is u ra d y k a łó w z w ią z a n y c h z tymi p o d zb io ra m i.
