Some Characterizations Of Commutative Graded Perfect Rings

Let G be a commutative group and let R=??G(?)R? be a commutative G-graded ring.This thesis shows some equivalent characterizations of graded semiperfect rings and graded perfect rings.Firstly,it is shown that every finitely generated graded module over a graded local ring has a graded projective cover.Secondly,we prove that R is graded semiperfect if and only if R is a direct product of a finite number of graded local rings.Then we get some equivalent characterizations of graded perfect rings,for example,if R is graded perfect if and only if R/Jg(R)is graded semisimple and every nonzero graded module has a gr-maximal submodule,if and only if every graded R-module satisfies the descending chain condition on cyclic submodules.At last,we prove that if R is a strongly graded ring,then R is a graded perfect ring if and only if Re is a perfect ring,and then R is graded Artinian if and only if R is both graded coherent and graded perfect.