Socles Of Group-Graded Rings And Modules

The main purpose of ring theory and module theory is to characterize the structure and properties of rings and modules. The socles and Jacobson radicals are important tools for characterizing the structure and properties of rings and modules. The socles and Jacobson radicals of rings and modules have been studied systematically and deeply. In this paper we devoted to study the socles and graded socles of group-graded rings and modules. We obtain some results dual to the Jacobson radicals of rings and modules and generalize some results about crossed products.Chapter two is the subject of this paper, in this chapter die main results about socles of crossed products in [3] written by Yi Zhong and Cheng Fuchang are generalized to group-graded rings. We give some concrete charaterization for the socles of group-graded rings. In particular, for afinitegroup G and a strongly G-gradedring ,wehaveTheorem 2.5 Let G be afinite group, let R be a strongly graded ring of type G and let M be a right R -module. Then we have(i) soc(MRi) is a right R -submodule of MR and soc(MR ) c soc(MR ) ; (ii) soc(M) = soc(MR) if and onfy if for each R -submodule NR of MR , NRsemisimple implies NR semisimple; (Hi) if M has no \G-torsion, then soc(MK ) = soc(MK).Theorem 2. 7 Let G be a finite group Jet R be a strongly graded ring of type G. Then soc(Re ) is a G -invariant ideal of Re.Theorem 2. 8 Let G be a finite group Jet R be a strongly graded ring of type G. Then we have if and only if for each simple right ideal Lof R , R is a semisimple right ideal of R ; (in) if Re has no \G\ -torsion, then soc(RR) = soc(Re )R .Theorem 2.10 Let G be a finite group , let R be a strongly graded ring of type G. ThenIn chapter ihree, in order to study the socles completely we discuss the graded socles. Making use of the properties of the socles, Jacobson radicals and graded Jacobson radicals of group-graded modules, we get some concrete characterization for graded socles of group-graded rings and modules. We discuss the relationship between socles and graded socles of group-graded rings and obtainTheorem 3. 9 Let G be afinite group, let R be a strongly gradedring of'type G .Then we haveCorol lary 3.10 Let G be afinite group, let R be a strongly graded ring of type G. Then...