TRU Mathematics 21−2 （1985）
NσrES ON GOING−UP THEOREM AND GOING−DOIVN THEOR日M
ON DIFFERENTIAL IDEALS
Mamoru FURUYA
（Received October 29， 1985〕 0． INTRODUCTION The lying−over theorem on different ial ideals with respect to a higher derivation of infinite ranl（is proved by S・Sato ［9］ under the assumption that the cons iderable ring is NOetherian． In the paper ［2， Theorem 1］，W．C．Brown proved the theorem without this assumption． Namely： Assume that a ring 1∼’is integral over its sUbring R．1．et 4ニ r（iy be a higher derivation of rank oo of R’which restricts to a higher derivation of raIハ］（oo of R． Suppose P is a g−differential prime ideal of R． Then there exists ad−differential prime ideal P’of R t such that P’n R＝P． From the theorem， he proved the going−up theorem and the going−down theorem （under the assumption that the cons iderable ring is Noetherian 〕 on differential ideals with respect to a higher derivation of infinite ran］k． In this paper， we first show a following existence theorem・ THEOREM （1．7）． Let 5 be a multiplicatively closed subset of a ring R and let 4＝ rdi／ be a higher derivation of rank m r∼η ≦ ◎oノ． If a d−differential idea1 ヱdoes not meet 5， then there is a maxima1 〈《−differential ideal M with respect to 5 such that〃contains τ． If m ＜o。， then such an idea1〃is primary． If m＝oo， then such an idea1 〃 is prime． Next we define a maximal d−differential P−primary ideal of a ring R for a prime ideal P and for a higher derivation 40f finite rank・From the theorem （1．7） and the definition， we prove the lying−over theorem， the going一町）theorem and the going−down theorem on differential ideals with respect to a higher derivation of finite rank． Lastly we get a detailed theorem more than the Brown，s lying−over theoremmd prow it by veワsi㎎1e㎜er． Namely：
THEOREM 〔3．2）． Assume that a ring J∼’is integral over its subring R． Let dニ「ちノbe a higher deriv・ti・n・f・ank…f R t・hi・h re・t・i・t・t・ahigher derivation of ran］k o。 of R． Suppose P is a d−differential prime ideal of∼ぞ． Then there are prime ideals of R’which lie over P；these prime ideals of R・231
are a］・14〒di．ffere口tia工↓．： し ㌧．．． ’ ： ’i． ．・：．”．’ ．『1． i ．，1 町㎜ng use°f・唾．the°「r1，（5・3）・、順，pr・W・ph・g・i・g一叩th・・re−d・he going−down theorem （without the assumption that the consideral）1e ring is Nbetherian） on differential ideals with respect to a higher derivation of infinite rank． 1． PRELIMINARIES AND EXISTENCE THEOREM AIl rings in this paper are assurned to be cormtative with a． unit element． Let R be a ring． A derivation of R is all additive endomo’rphism d！ごR→Rsuch that（i ra●ノ・ニad Cb）・＋d（aJb for everyα，あ ∈ R． The set層of a11．deTivations of R is denoted by Der （RJ． 1、et m be a posit．ive integer． A higher derivation of ranl（ m°fR． is．・・eqU・n・e・dニ「d。・d、・・∴・∂mノ・f・dditive・nd・m。rPhi・m・ di ・R→R such that： ・ ．、 ． ‘ ． ・
（1〕dO i・’th・id㎝tity㎜P・ 1．・・． ・層．
（2）d・「ab？ ＝’Xi・」己ω4〆わノf位eve「y・・「2≦・≦m） and for・e・・ry・・わ．・R・ −lt f・11・）w・that dl i・ad・・ivati・n・f R・：Tfl・．set・f al1 highe・deri・・ti・n・・f rank m of 1∼is denoted by eDerM rRノ∴Ahigher derivation of rank。。 of R is an i・fi・ite・equence d・（d。・d、・d2・…ノ・u・h t｝r・t r％・∂、・…．・d。）∈肋・・栩rR／fb・ every m≧ ゴ・ The set of all higher derivations of ran］k’●o of∼∼．is denoted by エ （R）．” ・ ．， ． ・ ． ・ 一 、 ． ．』 ’ HDer． L・tfば・・R’be a㎞・。・・rPhi…f・i・g・・L・t 4・rdi）．・∈・HD・Mm rR／rm…ノ ’and・・t 4’・イ∂膓ノ・・HD・幽・り・u・h l・ha・fdi．・dl・f ’f・r a・・i・．th・h…ha…ay that d’is an extension of d and d is a Testriction of．∂’． ’ Let 5 be a．mlltiplicatively closed sUbset of R． Then．it．is we11㎞om’ that every higher derivation of ran1（η76m ≦−co） of 1∼can be諏iquly∈…xtendedt°a higher deri・・ti…f・anlc m・f RS・ ’ ．1
1、et．dニ （d ．） ∈ el）erM （R／ rη7≦ooノ：and．1et τbe an ideal of R．・’lheri’we sha11 − zsay t｝誕工i・d−differ・・ti・1．if∂￠ω⊂τ f・r ・11 i・ ・ ． ．
The fbllowing lemma can be easily Pfoved from the・definiti6n．’ ‘ ㎜〔L1）・五・t f・R→R’b・a h・m・m・rphi・m・・f吻9・．−’丑・t d・∈HD・ノ∼倒佃≦b・」 an∂z・t d’∈肋・r用rRりbe’・an・｛1伽8鋤・ア．dド鬼・εh・餌励w
prope？t乞es ape satisfie∂． ．’・ ご．・ ． ’ ． 1’．．傷 rヱノ・．1アτis’α4−differθntt’aZ ideaZ of J？， thenヱR，． is a’dL（江∫アθ㌘eカtiaZ idedZ of．R’，・whθ，eヱ｝∼才is the’idb’aZ genevαカθ∂・by．了r刀in R，≧ ‘2）Z｝et 1●θ⑳z ideaZ oヂ」∼・．The力刀守？i8 d．㌔とtifferent’iaZザ・a批d onZy’￠fτ1rmり． z84−∂ifferentinz． ・：・ ・ ．． ・．・ ・ 「
r3／L・tτb・α・i・le・z・f肋乃励・ati・fi・sτ・f−i riRり． th。nτi． d− difアlarentiaZ if and onZy if IR， is d’−difアe㌘entiaZ．233
DI FFERENTIAL IDEALS r4把げz’ difアθアθntiαZ： i・αd・−di∬・renti・z・id・αz・了R・， th。。子1 rエりas d一 L日嚇1A 〔1．2）．1｝et王Z）eαn i（ieαZ of R and’Zet∂； rd，ノ ∈班）er°°rRJ sueh thαt − z ちω⊂z「i＝o・ヱ・…・・−1・n≧力・P・t d。「x）・u．r・∈f）・Th・n f…n〃 positiv￠ integθP Z， we get that ．d。、…㍉・みy、・％・∫・． PROOE I．et E be the automorphism of R［［司］ associated to 〈i 〔cf． ［4］ or ll］；：霊ぽ惣、E三1；。。芦≠d・「・・ZZ’ ’ d・ 「・Z／・2・…；・・h・d・fi・・…n 託1。㍗撰1；2．諭ノz ≡uZtnZ≠．．． mod f［［t］］． ［［h・・d。zrxり・uz m・d・． PR・POS・T・㎝（1・3〕・L・t・R b・α・殉弓・rちノ∈ffD・・°°rRノ…d・Z・t・r b・α4−di汀erentiaZ ideaZ・行・Then砺rα輪αZ r rzノ・f l isαZS・d−differen．協Z．
PROOF． Assume that di rr rl〃 ⊂r Cl） r i二〇，ヱ，＿．sn一ヱ， n ≧ 1 ノ． For any＝，㌶狸lu；2「1鴇el謡lr元゜le蕊et隠：1。Zu
∈Tr一τノ． 『 In the following we use the notations of ［7］．Thus the assunrption°’the ideal Z has a．prilnary decomposition°＄is not necessary． PROPOSITION （1．4）．Z｝etτZ）θan ZdeαZ o王α？ing R， PαminimαZ prime 鋤・…f」・Q th・P励・・y・・ap・nent・fエわ・Z．・ngi・g t・P・・□∈・ID・rn’ （Rノ．伽 ≦o。ノ．Asszeηe Z is d−difアθ？θntiaZ． Thθn： rヱノif・m＝・・，亡乃・・P・・d Q・r・4−∂切 br・励・z， r2ノ 句〔m＜oo， t ze’n Q is（1−〈乏ifアθz・〈？ntiaZ．「 PR・OF・L・・d’・llD・・M（Rpノ b・the ex・・n・i・n・f・d・ Tl・・n rR。 i・d’− differ6・・i・・anq・hu・Q・・R，・Ri・d−differen・1・・by〔1・1〕・・f・・．・…h・n the radical P of Q．is also 4−differential by 〔1．3〕． PropOsition （1．4） gives us a sufficient condition for a ring R to be an integral domain． Namely：COROLLARY 〔1．5）． 五θヵ R he α ring and Zet d ∈ 班）e？c° （R）． Suppose αny prime ideαz r≠roノαnd R／ts not∂−diffe？entiαZ． Then R isαn integ？αz domain． PROOF． Let P be a皿inimal prime divisor of rOノ． Then P is 4−differential by （1．4）．Thus Pニ roノ． COROLLARY （1．6）． αny P？乞mavey ideaZ r ≠ ideaZ．
L・tR・beαrw・nd z・品c肥・♂rRノ伽…ノ．⑭ρ・sθ
〔0ノαnd R／is not d−differentiαZ． Then ro／isαprimary PROOF． Let P be a minimal prime divisor of rの and let Q be the primary component of rO／ belonging to P． Then Q is d−differential by 〔1．4）． Thus （g＝ roノ． We now．prove the following existence theorem． THEOREM 〔1．7）．Let 5ゐe α nuZtゼpZi已αtiVe ly cZosed suZ）set ofα ring Rαndz・垣c∼の・打Rノ伽≦・・ノ．1アα4一碗庁賠編・z碗αZz∂・・s・・t meθt s，砺・
there isαmαximaZ d−di∬erential ideaZ M tuith？espeot to 5 such that M⊃z． ff m＜。・， th・n sueh an ideaZ M is prima・・y・万m；r，輪・su・hαηideαz M is ppzme・ PROOF． Let F be the set of（1−differential ideals B contaning X such that B∩ 5＝empty． Then F is an inductive set （ the order being given by the inclusion relation）．Hence， Zorパs le㎜a i皿plies the existence of〃． Let d’C 肋・㍗rR。）b・the ex・・n・i・n・f口d・・t f・R→RS b・th・・can。・ica・1：1㌶蕊麟：。i’；dl㌻二1，ll，（；’1跳1㌶；ei：xlts a d’
differ。nti。1 by〔1．1〕．（in・th・・th・・h・nd子ヱωi・ap・imary id・a1・u・h th・t 万1rθノ・M、nd了一ヱreノ∩S・・叩ty． th・・了一2 rQ）∈F・nd theref・re・M・f一ヱrQノ． Hence M is primary． If m＝。。， then the radical r rM） of M is 4−differential by 〔1．3）．Therefore r rM）∈Fand hence Mニr rM） is a pri皿e idea1． REト仏RK 〔1．8）． 〔1〕 If R is a Noetherian ring， then any d−differential ideal of R can be written as an intersection of d−・differential． primary ideals． Furthermore if m＝。。， then any prime divisor of a 4−differential ideal is also d−differentia1． Therefore Theorem 〔1．7〕． is obvious （cf． ［3］，［10］）． （2） If m＜。o， then the primary ideal M of （1．7） is not necessarily prime． To see this， we consider the following example：235
DIFFERENTIAL IDEALS
EXAMPLE． Let Rニk［x］be a polynomial ring over a field k of characteris− 2 ticρ≠ 0． Define Z）∈Z）er （R） by Z）rX） ：＝ヱ and Z）（k） ＝ 0． Cons iderヱ・＝XρRand 5＝R−XR． Then 5 is a multiplicatively closed subset of R， エ is a D− differential primary ideal and l ∩ 5 ニ empty． Furthermore 〃ニxρR is a皿aximal Z）−differential ideal with respect to 5 such that M⊃エ． 2． GOING−UP AN［） GOING−DOWN I In this section，？ηis always a positive integer．1．et R be a ring， P a prime ideal of R and let Q be a primary ideal belonging to P． 1、et （f∈ffl）erM （R）． Then we shall say that（g is a maxima1 4−differential P−primalγideal if the following conditions are satisfied： （1〕 Q is d−differentia1． 〔2〕 If Q’is a d−differential primary ideal such that Q⊂Q’⊂P， then Q ；ρ’． ’ The following properties are easily proved from the definition and （1．7）． 〔1） If P is d−differential， then P is maximal g−differential P−primary． 〔2〕 If Q is d−differentia1， then there exists a maximal d−differential P− primary ideal Q「such that （2⊂ρ’⊂P． The ideal Q「is unique． 〔3） If（〕is maxilnal d−differential P−primary and ifエis a d−differential ideal such that Q⊂f⊂P， then Q ＝一τ． PROPOSITION 〔2．1〕． Z｝et王 ：R→R’わθ α homomorlphism o王riアτσ5， d∈ ED・㍗Rノ・nd・d t∈eD・rM CR’ノ・n・苫協・i…f4． S・PP・・θの・・m・xim・Z d− di方eceentiaZ P−primαry ideaZ of R． Then there emistsαd’−diffe？entiαz P？imαryideaZρ’of R’Zying OVeアρ芽αnd onZy分アーヱ（eRり＝Q．
PROOF． If万Z rQRり・e，1・t 5 b・th・im・g・・f R−Pi・R・． lh・n eR’d… not meet 5 and QR’is（1「−differential by 〔1．1〕．Therefore there exists a（f「− differential primary ideal Q’of R， such that Q’⊃QR’and Q’∩ 5 ＝ empty by （1・・〕・恥・th・rm・re f−1 rQり・・Q、 i・9−diff…n・i・・by（1・1）a・d・Q・Q、・P・ 伽・w・g・te・Qヱ・The c・nv・rse i・・bvi・u・・ THEOREM 〔2．2〕．∠lssz〃ηθ thαtαriアτg R’is integ㌘αZ OVer its su力ring R・ Z｝et9∈日D♂側whi・h rest？i・カ・t・α励h・？ d・nivαti…f rαnk m・了R．5塑ρ・SθQ
isαmαりσ仇αz d−diffeアentiαZ P二ρrin∼αry ideαz of R．助θηthevee exist合αd− difi「θTentiaZ primαrly i（feαZ Q， o了R’such that Q，∩ RニQ． （Lying−over Theorem）PROOF・mtぽ∩R＝Qヱ・th・n w・g・t Q⊂el⊂PR’∩R＝Pby［1・（5・10）］・
T？・u・e⊂eヱ⊂P・Fu・th・rm…Qヱi・d−differenti・1 by〔1・1）・Theref・・e Q＝Qland hence QR f∩Rニθ． Therefore by （2．1），we have a 4−differential primary
ideal Q’such that Q’nR＝Q．
THEOREM 〔2．3）． ntth the sαητθ R， 」∼， and d αs in Theorem r2．2ノ， fo？ αn召 d− diff…nti・Zρぬ・・y id・αZ・e2・Q2・f R・鋤微t Qヱ⊂Q2・・nd・f・・α・四一 differentiαZρrimα？y ideaZ ei of R’Zη仇g OVer Ql・the？e existsα4−dtfferen−ti・Z p励・均鋤αZ％・f R’Z⑳θ・V・？ e2・鋤微t Qi⊂Qi if Q2 i・m・xim・Z
d−diff・・enti・Z P2−P・im・・〃・・h… P2 is th・・α伽・Z・f Q2・〔G・ing一叩加・。rem〕 PROOF・L・t Pヱ・nd弓b・th・radica1・f a、 and Qi・・espectively・Then we9・tPヱc P2㎝d弓∩R・P2・Th・・th・・e exi・t・ap・im・ideal pS°f R’1ying
・ver P2・u・h th・t弓⊂pi by［7・（10・7〕］・P・t f’＝Qi≠e2R「・1hen 「’is 9’− diff・r・nti・1・nd pS⊃∫’・Thu・there exi・t・ag−differenti・1 P・i皿ary ideal QS 。f R’・u・h th・t∫’⊂％⊂pi・恥t錫∩R＝Q・〕th・n Q i・d−diff・rential and Q2⊂Q⊂P2・Th・・e2ニa・
COROLLARY 〔2．4〕． Wtth the same R αn（！Rlαs in Theo？em ↓2・5ノ， Zet Z）∈ D・・「Rり・ueh th・t D「Rノ⊂R・F・・ any D−dtff…nti・Z P・im・・y id・αZ・ρ1・Q2・f R・励微力ρ1⊂02・・nd・60r any D−diff…nti・Z・P・tm・・召鋤αz弓゜f R’Zgw
・v・・ Ql・吻…x’i・t・aD−diff…nta・Zρ吻・ry ide・Z㌶・f Rr Zη働・V・r Q2・鋤吻力鰐⊂％if Q2 ts m・ximαZ D−diff…nti・1 P2一ρ励・ry・wh…P2 is the
radieαz・f Q2・ PROOF． Since r・，Dノ∈eD。。i （Rノ， the c。・・11・・y f・11・w・f・。m（2．3〕． THEOREM （2．5）．Let Rゐθα normaZ ring and Zet R’わθα「tフzg sueh thαt r力 R⊂R’，r2／R’is integpaZ over Rαnd r3／no non−zeアO eZement of R tsαzeveo divis。P in R’． Let d∈HD・TM （Rりwhi・h restri・t・t・α励力・ア±仇α力鋤・f ？α・km・fR・F・P any d−diffeT・nti・Z P・iM・id・αZ・Pヱ・P2・f R・鋤撫力Pヱ⊃P2・・nd
鋤α・四一diff…nti・Z・P・tm・id・aZ pi・f R’z〃w・…Pヱ・th… ・xi・t・ad−
diff・蹴励Zρ・imα・y id・αZ％・f・R’ Zy物・v・・P2・u・h微力弓⊃％・（G・ing−
down Theorem） PROOF・L・t P’ b・ a minim・1 P・im・di・i…。f P2R t・u・h th・t弓⊃P’・th・n P’∩R・P2 by［5・（5・B〕］・L・t％b・th・P・imary・・mp。…t・f P2R’ b・1・ngi・gt・pt・th・n％i・4“diff・・enti・1 by（1・4）and P’∩R⊃弓∩R⊃P2R’∩R⊃P2・
Theref・re w・g・t％∩R・P2・nd弓⊃％・
REMARK （2．6〕． In Theorem 〔2．2〕，the ideal Q「is not necessarily primeDIFFERENTIAL IDEAI．S
237
even ifρ is a pri皿1e idea1 〔see Exaiql）1e of Remark 〔3．7） ）． 3． GOING−UP AND GOING−DOWN II In this section we treat a higher derivation of ra皿k◎。． We shall prove the following proposition・ PROPOSITION 〔3．1）． Let王 ：R→R，力θαhomomorlphism o了rings， 《1∈ ffD・・°°窒q）・・口’∈eD・・°°rRり・・ぬ…鋤・f 4． S・PP・SθP乞・α凪苛…nti・Z
prime ideαz of R・Then there exi合tsα4LdifferentiαZ prime ideαz P’o王R’Zン仇gOVe？P分αnd onZy if万ヱ↓PRリ＝P．
PROOF． lfアーヱrPRり・P，1。t 5 b。 th。 i皿。g6。f R−Pi。 R’． th。n・PR・ d。es not meet s and PR’is d’−differential by 〔1．1〕．Therefore there exists a d，− differential prime idea11）いof R， such that P’⊃PR’and P’∩ 5＝e町）ty by 〔1．7〕．A・d・w・g・t子1rPり・P． Th・・c・一・e f。11㎝， by［1，（3．16〕］． i∼re now prove a detailed theorem more than the Brown’s lying−over theorem by very simple ma皿er． THEOREM （3．2）．Asszune thatαring R’is integrαZ over its suhring R． Let d∈ffD・・°°rRりt・hi・h ”est？i・t・カ・α殉』4・rivati・n・ア㌘α泳…芦． S・PP・εθP is α4−d乞ffθrθntiαZ P？iηe 乞dθaZ of R． Thθn the？e a？e Pアimθ ZdeαZS of R「whioh Zie OりθアP《 these P？ime tdeaZS O王Rtα？e αZZ 4−di∬erentiαZ．〔Lying−over Theorem） ． PROOF． By ［7， 〔10．8〕］，there are prime ideals of R， which lie over P． Let P’be any one of these prime ideals． Then P’is a minimal prime divisor of PR’． Since PR， is g−differentia1， P「is 4−differential by （1．4〕． The following （3．3） and 〔3．4） are easily proved fro皿Theorem （3．2〕． COROI．LARY 〔3．3）． Mth the εα7η（？R， R’and d α8 in Theorem r3．2ノ， Tb？ αny9−diff・・enti・Zρ励・td・αZ・P1・P2・f R・励吻力Pコ⊂P2・・雇鋤α・鴎一
diff・”・nti・Z P・im・id・αz pi・了R「z〃鋤・…P、・th・・eθ・ci・t・αP痂・id・αz PS ・fR’Zy物・v・p P2・uch th・方弓⊂PSS thi・P・ime id・αZ PS i・4−diff・・enti・Z・ （Going−up Theorem〕 PROOF・The existence of the prime ideal P》 follows from・［7・ （10・9〕］・And P》i・9−diff・r・nti・1 by〔3・2）・COROL工ARY （3．4）． ntth the sαme R αnd R「αs in Theorem r2．52， Zet 4 ∈ HDe？°°窒qリwhich rest？i・古S t・αhighe？de？鋤ti・n・了rank。。・行． F・㌘αη四一
