A characterization of gr-simple rings is given by using the notion of componentwise-dense subrings of a full matrix ring over a division ring. As a consequence, any G-graded full matrix ring over a division ring is isomorphic to a dense subring of a full matrix ring with a good G-grading. Some conditions for a grading of a full matrix ring to be isomorphic to a good one are given, which generalize some results in: Dascascu, S., Lon, B., Nastasescu, C. and Montes, J. R., Group gradings on full matrix rings, J. Algebra, 220(1999), 709-728.
For a commtative ring R and an injective cogenerator E in the category of R-modules, we characterize QF rings, IF rings and semihereditary rings by using the properties of the dual modules with respect to E.
In this paper we introduce the notion of relative syzygy modules. We then study the extension closure of the category of modules consisting of relative syzygy modules (resp. relative к-torsionfree modules).
The structure of rightF-⊥ T-approximations of any finitely generated module over an artin algebra Λ is given, relative to an additive subbifunctorF of Ext Λ 1 (-, -) and anF-cotilting moduleT.
