12035

11030

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1395

هشت، [70] ص.: مصور

ص. ع. به فارسي و انگليسي

واژه نامه

1395/11/05

كتابنامه

علوم رياضي

رياضي

ID11030

The dual notion of the prime radical of a module Niloofar Kianersi n kianersi@math iut ac ir 2017 Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 Iran Supervisor Dr Mohammad Reza Vedadi mrvedadi@cc iut ac ir Advisor Dr Mahmood Behboodi m behboodi@cc iut ac ir 2010 MSC 16D50 16D60 16D80 16N20 16N60 Keywords Second submodule Second radicals Prime submodule Prime radical Socle of a module AbstractLet R be an arbitrary ring By a proper submodule N of a non zero right R module M we mean asubmodule N with N M A right R module M will be called a second module provided M 0and annR M annR M N for every proper submodule N of M By a second submodule of amodule we mean a submodule which is also a second module A non zero R module M is called aprime module if annR M annR K for every non zero sub module K of M A proper submoduleN of a module M is called a prime submodule of M if M N is a prime module The intersection of allprime submodules of a module M is called the prime radical of M and is denoted by rad M If thereis no prime submodule in M then rad M M The prime radical has been investigated by variousauthors over the past 20 years The second radical Sec M of a module M is de ned to be the sumof all the second submodules of M we prove that if M is any right R module over such a ring thenSec M T L for some semisimple submodule T and submodule L such that L Rad L whereRad L denotes the Jacobson radical of L We de ne the notion of m system set which is the dualnotion of m system set Let R be a ring and M be a right R module A subset S M 0 is calledan m system if for each ideal A of R and for all submodules K L M annK L A S M and K L A S M imply that K L S M For a submodule N of a module M we de ne N x N there is an m system S such that x S and N S M s If there is no second submodule of N then we puts s N 0 It is proved thats s N Sec N wedetermine the second radical in some cases Let R be a ring such that R P is right artinian for everyright primitive ideal P If M is a noetherian right R module we determine all the maximal second

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