توصيفگر ها :
دوگان-مربع آزاد , توزيعي , تماما پايا , تميز , ددكيند متناهي , خاصيت تبادل , شبه دئو
چكيده انگليسي :
Dual square free modules Hossein Alikhani october 20 2020 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 8311 IranSupervisor Dr MohammadReza Vedadi mrvedadi@cc iut ac irAdvisor Dr Mahmood Behboodi mbehbood@cc iut ac ir2000 MSC 16D40 16D50 16D60Keywords Dual square free distributive fully invariant clean Dedekind finite exchange property quasi duoAbstract This M Sc thesis is based on the following paper Dual square free modules Communications in Algebra 47 7 2019 Y I M Y 2954 2966 This study seeks to investigate dual square free DSF modules Module M is called DSF if M has no proper sub modules of A and B with M A B and M M A BEvery summand of DSF sub module is a DSF module Furthermore this set of modules is closed under homomorphicimages Module M is distributive iff each sub module of M is DSF DSF maximal sub modules are entirely fully invariant Particularly a ring R is a right DSF R module iff R is a rightquasi duo i e any maximal right ideals are two sided On the other side every DSF module is Dedekind finite If M possesses the finite exchange property then M has the finite property and EndR M ring has stable 1 range Ultimately it is demonstrated that every DSF module like M possesses the finite exchange property iff M is cleaniff M has the full exchange property Drawing on the results of the study what comes next is worth mentioning First the following conditions for R module M are equivalent 1 M is DSF 2 If L is a factor module of M such that L N N for some module N then N 0 3 If M A B where A B M then M A and M B are factor orthogonal 4 If M A B where A B M then Epi A A B B A B 0 5 If M A B where A B M then Iso A A B B A B 0 Second the subsequent proposition provides an effective mechanism for raising typical examples of DSF Given that M A B considered as a DSF module and B n Si is a finite direct sum of non isomorphic simple i 1modules Then M is a DSF module if and only if A and B are factor orthogonal Another crucial outcome of the study demonstrates that the following conditions for ring R are equivalent 1 A ring R is right DSF 2 R is right quasi duo 3 R is simply distinct as a right R module Next we show in a proposition that if M A B is a DSF module and f A B be an R homomorphism thenImf is small in B Imf B that results M is a DSF module with Rad M 0 if M A B then HomR A B HomR B A 0 by means of DSF results we also show that following conditions are equivalent on ring R 1 R is a right quasi duo and a B ring 2 J R is right T nilpotent and R J R is strongly regular As another outcome of this study we mentioned that if M is a DSF module then every epimorphism f M Mhas a small kernel Kerf M and also
