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چكيده انگليسي :

Second Modules over Noncommutative Rings Aliasghar Zarean aa zarean@math iut ac ir 2014 Master of Science Thesis in Farsi Departement of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranSupervisor Dr Atefeh Ghorbani a ghorbani@cc iut ac irAdvisor Dr Mohammad Reza vedadi mrvedadi@cc iut ac ir2000 MSC 16D10 16N60 16L30 16L60 16L99 Keywords Attached prime ideal Hollow dimension Second module Semilocal rings Abstract This thesis is based on the article Second Modules over Noncommutative Rings written by S Ceken M Alkan and P F Smith The notion of second modules and second submodules introduced by Yassemi which is a dual to thenotion of prime modules Let R be an arbitrary ring A nonzero unital right R module M is called a second moduleif M and all its nonzero homomorphic images have the same annihilator in R A nonzero unital right R module Mis called a prime module if M and all its nonzero submoduls have the same annihilator in R By a prime submoduleof a right R module M we mean a submodule N such that the module M N is prime By a second submoduleof a module we mean a submodule which is also a second module Let R be a commutative ring and let M be anonzero R module Given any element r R let r M M denote the endomorphism of M defined by r m rm m M It is easy to check that M is prime if and only if for each r R either r is zero ora monomorphism In other words M is prime if and only if for any r in R and m in M rm 0 implies thatm 0 or rM 0 On the other hand the R module M is second if and only if for each r R either ris zero or an epimorphism Putting it another way M is second if and only if for any r in R either rM 0 orrM M Note that if R is any ring and M is a second right R module then P annR M is a prime idealof R and In this case it will be convenient to call M a P second module It is proved that if R is a ring such thatR P is a left bounded left Goldie ring for every prime ideal P of R then a right R module M is a second moduleif and only if Q annR M is a prime ideal of R and M is a divisible right R Q module Let R be any ringsuch that for each a R the ideal RaR is finitely generated as a left ideal let P be prime ideal of R and let aMi i I be any collection of P second right R modules Then the right R module i I Mi is P second LetP be a prime ideal of a ring R and let M be a P second right R module Then every nonzero supplement in M isa P second module It is shown that if R be a ring and let Ni i I be a chain of second submodules of a rightR module M Then N i I Ni is a second submodule of M If a ring R satisfies the ascending chain conditionon two sided ideals then every nonzero R module has a nonzero homomorphic image which is a second module Every nonzero Artinian module contains second submodules and there are only a finite number of maximal membersin the collection of second submodules If R is a ring and M is a nonzero right R module such that M contains aproper submodule N with M N a second module and M has finite hollow dimension n for some positive integern then there exist a positive integer k n and prime ideals Pi 1 i k such that if L is a proper submoduleof M with M L a second module then M L has annihilator Pi for some 1 i k Let P be a prime ideal ofa ring R let n be a positive integer and let Li 1 i n be a coindependent family of submodules of M suchthat M Li is a P second module for each 1 i n Then M 1 i n Li is also a P second module Everysecond submodule of an Artinian module is a finite sum of hollow second submodules Finally we show that if Rbe any ring and M be any right R module which satisfies the AB5 condition and N be a submodule of M suchthat M N is P second for some prime ideal P of R Then there exists a submodule L of M minimal with respectto the properties that L N and M L is P second

زارعان، علي اصغر

مدول هاي دوم روي حلقه هاي تعويض ناپذير

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