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-Z E S -Z Y T Y NAUKOWE W Y Ż S -Z E J S-Z K O Ł Y PED AG O G IC-ZN EJ W B Y D G O SZ CZ Y P ro b le m y M a te m a ty c z n e 1 9 8 2 z .4 A n d rzej Nowicki UMK Toruń R y sz a rd Żuchow ski W S P B y d g o sz c z SO M E R E M A R K S ON S Y S T E M S OP IDEALS
If R is a commutative ring with Identity and M is a n y s e t of id e a ls of R , then a p a ir ( R , M) w ill be c a lle d a system of id e a ls , if the following con d ition s a r e sa tisfie d
A L R is an elem ent o f M,
A 2. A n in te rse c tio n of a n y s e t of elem ents of M i s an elem ent of M,
A 3. A union of a n y non-em pty s e t, to tally o rd e re d b y in clu sion , of elem ents of M is an elem ent o f M,
A 4 , T he null id e a l b elon g s to M,
A 5 . If A , В belong to M, then A+B b elon g s to M, Aft. If A , В b elon g to M, then A B b elon g s to M, A 7. If A , В b e lo n g to M, th en A :B b e lo n g s to M, w h e r e A :B - -< r € R{ A r * b € А L b € B J A 8. If A b e lo n g s to M, a n d x Is a n y e le m e n t of R , th en oo A - ( J (A :x n ) b e lo n g s to M. x n - о L et ( R , M ) b e a s y s te m of I d e a ls . E le m e n ts of M a r e c a lle d M - ld e a ls . If E i s a s u b s e t of R , th en w e d e n o te b y £ e^ th e s m a lle s t M—id e a l c o n ta in in g E . If A i s a n id e a l of R , then w e d e n o te b y А +ф th e g r e a t e s t M - id e a l c o n ta in e d in A . If ( R , M ) a n d ( S , N) a r e s y s t e m s of id e a ls , th en a rin g hom om orphism f:R — > S w ill b e c a lle d a m orphism of s y s t e m s if:
2. A n Ideal g e n e ra te d ln S b y the im age o f a n y M ^ d eal is an N-4deaL
In the p a p e rs |VJ , ]Vj , th e re a r e elem ents o f the th e o ry of s y ste m s o f id e a ls an d among o th e rs im portant e x a m p le s o f sy ste m s a s f.e. s y ste m s of differential id e a ls ( j_4j, jś] ) , s y s te m s o f differential id e a ls with re s p e c t to a h ig h er d erivatio n ( [2] , [э] ) and sy s te m s of
hom ogenous id e a ls in a ring with a grading ( j l j , ;»] ).
In this p ap er w e g ive som e rem a rk s on s y ste m s of id e a ls and w e d e s c rib e som e new ex am p les o f th o se .
PROPOSITION 1 . If M is a s e t o f id e a ls of R sa tisfy in g A 3 , then the condition A S i s eq u ivalen t with the follow ing condition:
(i) If A £ M, then A T £ M, fo r e v e r y m ultlplicatlvely s e t T ln R, w h ere A p » £ r 6 R; rt £ A , fo r som e t £ t|- .
PROOP. T he p ro p e rty (l) im plies A 8 , b e c a u s e T * ^ х П{ n - О, 1 , 2, . . I i s a m ultiplieatively s e t ln R an d • A ^ . Now, w e
p ro v e the in v e rs e im plication. Let A £ M an d le t T b e a m ultiplieatively s e t ln R. C o n sid e r a fam ily XI • |S| S i ł a mulUplicaUvely s e t in R, su c h that S с T and A ^ £ M f . The fam ily XX is non-em pty, b e c a u s e fo r t £ T , ^ tn, n - О, 1 , 2, . . . ^ is a m ultiplieatively s e t contained in T, and b y A 8 , w e h a v e A | - A t £ M. N otice, that Z sa tle fie e the assum ption o f Lemma K u ra to w sk i-Z o m . Let
1 £ I
j
be a chain in XI • T hen |A g • i £ Ij
is a chain in M and S m u S. is a m ultiplieatively s e t in R, contained inI 6 1
T . S in c e A= - M A , then, b y A 3 , we h a v e A - £ M. T hus
_ I £ I 1 S
S £ 5 1 .
Let S С T be a maximal elem ent in XX • T hus A s é- M. S u p p o se that S T , and le t t £ T \ S . T hen U - s j/ * } ls « m ultiplieatively s e t contained in T and p ro p e rly containing S . B y A 3 , we h a v e - A s |tnj . - (A s ) t € M. T h e re fo re S Ç V a n d U g £ , , in sp ite of S is a maximal elem ent in XX . S o , w e get S - T, a n d A T £ M.
PROPOSITION 2. U (R , M) U a a y a tam o f id e a ls , than an a lg a b ra ic a um o f a n y s e t of elem en ts o f M, i s a n elem ent o f M.
PROOF. If A la an Ideal of R, then the condition A £ M la eq u ivalen t with the Implication! x 6 A |xj d A ( s e e Щ ) . Let
[ A ,] , £ , be a co llectio n of M<4deals. If x ér 2 Z A j then x - a^ + . . ♦ a n b elo n g s to A . + . . . + A . , s o j V J С A . ♦ , . . ♦ A. с
I n I n
С Г а,.
A ssu m e now, that (к , m) is a s y ste m of Id eals, S i s a m ultiplieatively s e t in R and S - 1 R is a quotient ring of R with re s p e c t to S . Let N - { S - 1 A ; А С м | . It Is e a s y to p ro v e , that N i s the o n ly s e t of Id ea ls in S - 1 R s u c h that ( S - 1 R, N) i s a sy ste m o f Id eals and the n atu ral homomorphism ftR S - 1 R, r ,—» * is a morphism of sy ste m s. T he s e t N We s h a ll denote b y S - 1 M.
PROPOSITION 3. If S is a m ultiplieatively a e t In R and A € then
a ) (s a ) фф. - S “Ł ( ( A Ы b ) [ s ' V ] - s- ^ a]
PROOF. F irs t, w e p ro v e , that if A - A g , then ( S - 1a) ^ = S - 1 A
It U c le a r, that ( s “XA ) ^ - S - 1 B, w h ere В É Ы and В - В . H ence A ^ - ( A g ) ^ - ( Г 1 ( S ^ A ) ) - * - ^ ( ( S ^ A j ^ ) - Г * ( s ”1 В )
- B S - b. _
w h ere ft R —> S~ R is the natural homomorphism.
T h en w e h a v e S - 1 (A 1^) - S - 1 В - ( S - 1 A ) ^ , s o fin ally ( S - 1 A ) # - ( S _ 1 (A S ) ) ^ - s " l ( ( A s )1p . T h is e n d s the p roof o f a ) . T he p roof o f b ) i s stan d ard .
A sy ste m (R , M ) is c a lle d s p e c ia l. If the ra d ic a l of an a rb ita ry M -id eal Is a n M -ideal ( [V] ).
PROPOSITION 4. If S is a m ultiplicativaly s e t ln R, and (R.M ) Is a s p e c ia l sy ste m , then ( S - 1 R, S - 1 M) Is s p e c ia l too.
- 1 - 1
PROOF4. Let be a n y prime id e a l in S R. T hen О » S P, w h ere P ia a prime id e a l ln R d is jo in t from S . B y p roposition 3 a ) ,
*1 »
w e h a v e Q ^ - ( S P)^_ - s “ ( P ^ ). But (R , M) i s s p e c ia l, s o P ^ i s a prime id e a l o f R (T h .1 .2 [7] ) . S in c e P ^ . С P and TPoS-0
then P ^ Г» S m 0 F in a lly, GL ^ S '^ P ^ is a prime id e a l in S - 1 R and b y ^ 7]w e h a v e th e s is .
If P i s a prime id e a l in R, then w e denote b y (R p , Mp ) a system ( S ^ R , S _ 1 M ), w h ere S - R 4 P.
PROPOSITION 5. Let (R , M) b e a sy ste m o f id e a ls . The follow ing con d ition s a r e eq u ivalen t
( 1 ) (R, M) i s s p e c ia l,
( 2 ) (R p , 1* s p e c ia l, fo r e v e r y prime id e a l P in R.
(3 ) (r m • ) is s p e c ia l, fo r e v e r y maximal id e a l In R.
1 1
PROOF. T he im plication ( l ) =#• ( 2 ) fo llo w s b y P rop osition 4. It is c le a r, that (2)= ^ - ( 3 ) . W e p ro v e
(з)
“=^ ( l ) .C o n sid e r a n y prime id e a l P in R. W e sh o w , that P i s a prime id eal. Let be a maximal id e a l, s u c h that P С Let S ■ R v M^. T hen S hi p m ^6, and b y P rop osition 3 w e h a v e ( s ”^P) — S™^ ( P^)
B e c a u s e (S~^R, s “*M ) is s p e c ia l, s o is a prime id e a l in S - 1 R, h e n c e Р^ф. i s a prime id e a l in R.
Now w e g ive new exam p les o f sy s te m s o f id e a ls . F irs t, w e d e s c rib e a ll s y s te m s of Id eals in the rin g Z o f in te g e rs.
EXAMPLE 1 . Łet P - 1 p . , ...p . . . Д be a s e t Г i i
o f prime In teg e rs (finite o r in fin ite), and le t D « A p . l , p 2 2, . . . . p^n, . . be a s e t o f fixed p o w e rs o f elem en ts o f P . T hen (Z ,M p ]
w h ere MQ - | ( n ) ; n - ( p ^ l ) e l . . .( p ^ k ^ k , Sj ^ o j ^ |(0)| i s a system o f id e a ls . C o n v e rs e ly , e v e r y sy ste m (Z , M ) h a s the a b o v e form.
PROOF. It i s e a s y to p ro v e , that (Z , MQ ) is a sy ste m of id e a ls . W e sh o w , that a n y sy ste m of id e a ls (Z , M) h a s the form (Z , M p)» Let p
-jp;
p - a prime in te g e r, s u c h thatp|
n fo r som eИ Р - Р 2 , . . . , Р к , . . .
j
, then w e defineij - min |is there e x is te (n ) £' Ms n — p je , i > O, pj J ( с j , w h ere j - 1 ,2 ,... S e t D -
f
p * l, p2 2, . . ..p^k, . . . I . We p ro ve that M - M ^,le le
Notice, that if (n ) £ M, w h ere n - p ^ l . * ф иШ9 then b y the uniq u en eM of prim ary decom position and b y Th. 3 .4 Ls] , w e obtain (n ) - (p*^l) л
n . . . A ( p * s ) , w h ere ( p ^ l) £ M, . . „ , (p ^ *) € M. H ence, ai*o ( p * l) С M, . . . , ( p j ) £ M .
T h e r e f o r e M ^ с M. If (m) £ M, w h ere m - p ![ l. . then
£ M, fo r j - 1 , 2 , . . . , t. It i s o b v io u s that kj } ij. Let k^ - “ j i. + rj , w here О ■< r^ 4 ij, j - 1 , 2, . . . , t. U O, then (p jj) - (p]5) : (p'jVj) £ M^ w hich co n trad icts with minimality of ij. H ence, we h a v e (p^ j) - (p j^ )UJ £ MD fo r j - 1 , 2...t. F i n a l l y ( m ) - ( p ^ l ) П . . . П (pj^) € M D a n d M - M ß
We sh a ll d e s c rib e now, a ll s p e c ia l sy ste m s ln Z.
EXAMPLE 2. Let P be a s e t o f prime In tegers. T hen (Z , M _ ) ,
{ (n ) ; n - p , l . . .p^k; pJ t py .. . . i i
£ P , ^ ^ О... !k > ° } ^ { ^o )] i s a s p e c ia l system . C o n v e r s e ly , If (Z , M) is a s p e c ia l system , then th ere e x is ts a s e t P o f prime in te g e rs su c h that M - Mp .
PROOF. Is sim ilar to the proof of Exam ple 1 .
We c a n do the an alo g o u s d e sc rip tio n fo r p r in c ip a l Id eals dom ains. И (R , M) is a system o f id e a ls in R, then we denote b y M jx ] a s e t of id e a ls ift R [x] of the form A [x] - I а охП + • • • + a n* a , £ A j , w h ere A ą M. W e s h a ll, p ro v e , that (R [ x j , M [ x ] ) is a system of id e a ls . F ir s t , w e p ro v e two le m m as.
LEMMA 1 . Let f - a x n + . . . + a . g - b x™ + . . . . + b„n o m о belong to R [x] . If g. f - O, then ł - О
PROOP. S in c e g» f - ( 1 ) b m . a n - О <*> b m - l * n + b m * n - l - ° (m) b l « n ♦ b 2a r>_1 ♦ ...+ bn+ 1a e - О (m + l) bo a n + Ь1 а л _1 ♦ ... ♦ bna o - О 2 -n. -V л п n о
M ultiply the equation ( 2 ) b y a n, ( 3 ) b y a * , ... (m + l) by
V haV * S b m - l a n - ° I b nw2 n „ a 3 - O. ' m ' l a n b .a _ - О ,
“ o V * 1 - °
Ш41 H ence g-a n — O. L E M M A 2 . l e t f - а п хП + • • • • ♦ a 0 * 8 “ Ьт Х,П + * * ' + b 0 O*o .b elon g to R j j x j . . If g £ L J ( ° » )• then fo r e a c h p a ir i# J, w h ere k - 0
i ■ Ot 1 , 2. , . 9m , j m l # 2 ... n th ere e x is ts s ( i # j) £ N su c h
* * • • J ■
that b,. a p 1 • - 0.
PROOF. Induction with re g a rd of n, w h ere n - deg f. If n 0 , the Lemma is obvious» S u p p o se now, that Lemma is true fo r polynom ials f of d e g re e * < n, and fo r e v e r y polynom ial g. Let d eg f ш n, and g £ ( Ojf* ).
k - o
S u p p o se , that g. fP •> 0. Let h - f*3. A c o e ffic ie n t at the maximal p ow er of x in h is equalt a P - b S in c e g.h - 0, b y the Lemme 1 , w e h a v e g.bm+1 - 0 . T hen g . а Г - 0, w h ere r - p(m + l ) .
Let - f- a nx n, then g . £ +г - o, F in a lly g £ О (°И^), к —о w h e r e d e g Г, < r>-l, th e t h e s is fo llo w s b y th e In d u ctio n a ss u m p tio n .
C o r o lla r y , b e t A b e a n i d e a l in R a n d le t f - a ^ x „ £ • • • • + a 0> g - Ьт хГП + • • • • + bQ b e p o ly n o m ia ls in R £x]. If g e kU Q( A f x ] t f ) , th en fo r e v e r y p a ir i j , w h e r e i » 0 ,1 ,2 , . . . , m, j = 1 , 2, . . . , n, th e re e x i s t s s ( i , j ) e N s u c h th a t b ^ . a f ^ * A.
PRO O F. It s u f f ic e s to a p p ly uem m a 2 an d the is o m o r p h is m (% ) f x ]
THEO REM . L et f - а пхП + • • • • + a 0 ^ R Cx3 ond le t A b e and
id e a l of R . T h e n A [ x ] ^ = (A a П A& П . . . A^ ) Гх^
a o a l a n
1Й ’ (А { х ] : fk ) -
Л
( U ( A :a k ) ) [ x ] . .k * o i - о k - o
PRO O F. T h e in c lu s io n C: fo llo w s from C o r o lla r y , w e p ro v e the cm + . . . + b b e lo n g s to ( N ( A :a k 1 ö S * ' 1 l " 0 1 then b k a ^ o k , . . . , b ^ n k C A , S e t s k - m a x ( s . , . . . . , s n k ) . in v e r s e in c lu s io n . I f g - b x s s m + . . , + b o b e lo n g s to .* i - o |~0' ■ ( I )( A :a . i ) ) Гх~1 » i — ' then b k a Qo k , . . . , b ^ n k Ê A , S e t s k - m a x ( s Qk... .... s n k ) . fo r к - 1 , 2 , . . m. T h e n b ^ k E A an d g . fs 0 + sl + * + Sme A j x ] . E X A M PLE 5 . If ( R , m ) i s a s y s t e m s of i d e a ls , th en ( R IX) [x ] i s a s y s te m of i d e a l s too. P R O O F. It i s c l e a r , th a t th e c o n d itio n s A 1 - A 7 a r e s a t is f ie d . 3 y th e o b o v e T h e o re m , th e c o n d itio n A 8 i s a ls o s a t is f ie d .
EXA M PLE 4 . If ( R , M ) i s a s p e c ia l s y s te m of i d e a ls , then ( « Ы . M [ x ] ) i s a s p e c i a l s y s t e m of i d e a ls .
REFERENCES
[l] M.F. A tiy a h , I.G .M acdonald, Introduction to Commutative A lg e b ra Adi s on - W e s e le y P ublishing Com pany, M a s s a c h u s e tts 1 9 6 9 [2 ] W .C.Brown, W .K uan, Id eals and h ig h er d erivatio n in commutative
rin g s, C anadian Jo u rn al of M athem atics 2 4 ( l 9 7 2 ) . . [3] N.J a c o b s o n , L ectu re s in A b stra c t A lg e b ra v o l. II i I, D. Van
N ostrand Com pany. 1 9 6 4
[4] W .F .K eig h er, Prime d ifferential id e a ls in d iferen tial rin g s, Contribu tion to A lg e b ra , A C ollection of P a p e rs D edicated to E K olch in ,
1 9 7 7 , 2 3 9 - 2 4 9 .
[5 ] E .R .K olch in, D ifferential A lg e b ra and A lg e b ra ic G ro u p s, A cadem ic P r e s s . New Y ork, London, 1 9 7 3 .
[б] A.Nowicki, Prime id e a ls stru c tu re in additiwe c o n s e rv a tiv e sy ste m s (to a p p e a r) . .
[ 7] A.N owickl, R .Ż uchow ski, S p e c ia l sy ste m s of id e a ls i n commutative rin g s (to a p p e a r in. Com m entationes M athem aticae).
[8] O .Z ariski, P .Sam uel, Commutative A lg e b ra v o l . 2, Van N orstrand C.O, P rinceton . 1 9 6 0
[9] R .Ż uchow ski, S y ste m s of id e a ls in commutative rin g s (to a p p e a r in Com m entationes M athem aticae).
PEWNE UWAGI O SYSTE M ACH IDEAŁÓW STRESZCZEN IE
W p ra c a c h [7] , [9 ] podane s ą elem enty teorii system ó w ideałów i m iędzy innymi w a ż n ie jsz e p rz y k ła d y system ó w ja k np. s y s te m y ideałów
ró ż n ic z k o w y c h , s y s te m y ideałów niezm ienniczych ze w zg lęd u n a d e ry w a c ję w y ż s z ą i sy ste m y ideałów je d n o ro d n ych w p ie rśc ie n ia c h z g rad acją .
W n in ie jsz e j p r a c y podane s ą pew ne uwagi d o tyczą ce system ó w i o p isa n e s ą nowe p rz y k ła d y system ó w ideałów .
