关于半环的结构和同余

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1、则s／p=雪为S．／p。=＆的强分配格的充要条侔为P§o．第四章得出广义分式半环及其上的广义分式半模的一些性质，给出广义分式半环的泛性刻划，主要结论如下：定理4．5若R，A是含幺交换半环，设g：R—}A为半环同态，s，T为R的乘法闭子集，s￡T，且使9(s)为A的可逆元子集，9(T)为A的可消元子集，则存在唯一的同态h：筛1冗—+A使h，=9．关键词：半环，半环的强分配格，半环的强分配格对应的分配格同余，相等化子，同余可消元分类号：0152．73OnStructuresofSemiringsAndCong

2、ruencesonSemiringsLiuHongxingDepartmentofMathematics，ShandongNormalUniversityJinan，Shandong，250014，P．R．ChinaABSTRACTInthisdissertation，wegiveacharacterizationofringcongruencesonasemir—ing；besides，wediscusstherelationofasublatticeofthedirectproductofthelat

3、ticesofcongruencesonafamilyofsemiringsandasublatticeofthelatticeofcongruencesonthestrongdistributivelatticeofthosesemirings；finally，wediscussgeneralizedfractionalsemiringsandgeneralizedfractionalsemimodulesonthose．Themainresultsaregiveninfollow，InChapter1

4、，wegivetheintroductionandpreliminaries．InChapter2，wemainlygiveacharacterizationofringcongruencesonasemiringandgetthecharacterizationofringcongruencesonadditionalcommuta-tivesemirings．Theorem2．7LetRbeasemiring，TbeadensereflexiveideaofR，thenPTisaringcongrue

5、nceonRandT≤kerpT；Conversely，ifPisaringcongruenceonR，thenkerpisafulldensereflexiveunitaryideaofR，andP=PkerD．InChapter3，wecharacterizethecongruencesonastrongdistributivelatticeofsemiringsbythecongruencesonthosesemiringsandprovethatasub-latticeofthedirectpro

6、ductofthelatticesofcongruencesonthosesemiringsisisomorphictoasublatticeofthelatticeofcongruencesonthestrongdistributivelatticeofsemirings．Finally,wegetanecessaryandsufficientconditionforaquo-tientsemiringofastrongdistributivelatticeofsemiringstobeastrongd

7、istributivelatticeofquotientsemirings．Themainresultsaregiveninfollow．Lemma3．2LetS=，Pabeacongruenceon＆恤∈D)，and{风Io∈D)satisfyconditionVa，b∈&，(a，b)∈Pa—}V卢≥口，卢∈D，(o‰，口，bcpQ，卢)∈p卢，(A)4arelationP0nSisdefinedby(a，b)∈P，a∈&，6∈函{==争j7≥a+卢，(o。p。．，，6妒口．1)∈p

8、1．ThenPiscongruenceonS．Theorem3．13LetS=，andeachlpn，口beanisomorphism．Definemap妒：c台—÷￡c，where￡c={p∈￡slpsatisfiesconditiong)，{p。)卜_P，Then妒islatticeisomorphism．Theorem3．22LetS=，盯bethecorrespondingdis