資源描述:
《代數(shù)上局部?jī)缌銓?dǎo)子的性質(zhì)》由會(huì)員上傳分享,免費(fèi)在線閱讀,更多相關(guān)內(nèi)容在學(xué)術(shù)論文-天天文庫(kù)。
1、華中師范大學(xué)碩士學(xué)位論文代數(shù)上局部?jī)缌銓?dǎo)子的性質(zhì)姓名:沈如林申請(qǐng)學(xué)位級(jí)別:碩士專業(yè):基礎(chǔ)數(shù)學(xué)指導(dǎo)教師:譚志松;李桃生20040501⑨碩士學(xué)位論文MASTER’ST11ESIS摘要本文研究了特征為0的域上的代數(shù)的局部?jī)缌銓?dǎo)子,首先在第一部分給出了一般代數(shù)上的線性變換J。及c。的性質(zhì):(s。)2+04)2=I(這里的I是恒等映射1等,這也就刻畫了代數(shù)上的局部?jī)缌銓?dǎo)子的性質(zhì).第二部分考慮素代數(shù)上的局部?jī)缌銓?dǎo)子,證明了:若A是有單位元的素代數(shù),則A是無(wú)零因子代數(shù)當(dāng)且僅當(dāng)A上的冪零導(dǎo)子是0.之后考慮有限維素代數(shù)上的局部?jī)缌銓?dǎo)子,得到了導(dǎo)子成為冪零導(dǎo)子
2、的幾個(gè)充分條件:(1)設(shè)d是有限維素代數(shù)上的導(dǎo)子,若存在口≠0滿足碳口)=0,并且對(duì)于每個(gè)x∈爿,存在正整數(shù)n(x)J.fd礦∞“)=0,則d是冪零導(dǎo)子.(2)設(shè)dl,d是有限維素代數(shù)A上的導(dǎo)子,并且dl≠0,dId=ddl,如果任意x芒爿,存在正整數(shù)”@),使得dt礦∞0)=O,則d是冪零導(dǎo)子.(3)設(shè)dl,d2是有限維素代數(shù)上的導(dǎo)子,如果dl冼=d2dI,且荇對(duì)于每個(gè)元x都存在”0)和mO),使得函州叼虎忡’∞一O,則dI和噸至少有一個(gè)是冪零導(dǎo)予.最后還考慮特殊的多項(xiàng)式代數(shù)m】,并證明了其上的任一局部?jī)缌銓?dǎo)子為d=a。0(這是ao∈Fa是
3、15Ix]的求導(dǎo)運(yùn)算1.關(guān)鍵詞:導(dǎo)子;冪零導(dǎo)子;局部?jī)缌銓?dǎo)子:素代數(shù)AbstractInthispaper,weinvestigateintothelocallynilpotentderivationsofalgebrasoverthefieldFwithcharF=O.Firstly,wegivethecharacterization(s。)2+(c。)2=IoflineartranslationsS4andC“overgeneralalgebras(whereIisidentitymap、inthefirstpart,thatis,the
4、characterizationoflocallynilpotentderivationisshown.Inthesecondpart,westudylocallynilpotentderivationsofprimealgebrasandprovethat,iftheprimealgebraAhastheunit,theAhasnozerodivisorsifandonlyifeachnilpotentderivationofAis0.Afterthat,locallynilpotentderivationsareinvestigateda
5、ndseveralsufficientconditionsbecomingnilpotentderivationsaregiven,(1)Supposethatdisaderivationoffinite-dimensionallyprimealgebraA.If口≠0satisfiesd(口)=0andeachxEA3n(x)≥lsuchthat(I礦㈣=O,thendisnilpotent.(2)Supposethat函,darederivationsofthefinite—dimensionallyprimealgebraA,dl≠0a
6、nddld=ddt.IfVx∈A,3"(x)≥l,suchthatdid2"00(x)=0,thendisanilpotentone.(3)Supposedl,d2aleonesofthefinite—dimensionallyprimealgebraA,and函以=疵而.IfVxeA,Sn(x),聊@)≥1,suchthat礬“(x)d2”∽0)=O,thenoneof函and,/2isnilpotentatleast.Finally,weinvestigatethestructureoflocallynilpotentderivation
7、ofthepolynomialalgebra研胡;andtheformulaofeachlocallynilpotentoneisaoO(whereaoEF,0isthederivativeofFix]).Keywords:derivationnilpotentderivation;locallynilpotentderivationprimealgebra:U⑧碩士學(xué)位論文MASTER’STtlESIS§0引言眾所周知,在實(shí)(復(fù))數(shù)域上的函數(shù)的微積分理論可謂視為現(xiàn)代數(shù)學(xué)的開端.經(jīng)典的微積分理論已經(jīng)成為一種重要的數(shù)學(xué)工具,而其上用極限定義的微
8、分起著關(guān)鍵性作用.隨后,將微分加以抽象,相應(yīng)在環(huán),代數(shù)中就有了導(dǎo)子(或微分)的概念.在現(xiàn)代數(shù)學(xué)中的整分析,代數(shù)幾何及其代數(shù),導(dǎo)子也發(fā)揮著不可缺少的作用.例如,在二十