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 conse-quence, 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: Dáscáscu, S., Lon, B., Niǎstǎsescu, C. and Montes, J. R., Group gradingson full matrix rings, J. Algebra, 220(1999), 709-728.
In this paper we introduce the notion of relative syzygy modules. We then study the extensionclosure of the category of modules consisting of relative syzygy modules (resp. relative k-torsionfree modules).
The structure of right F-T-approximations of any finitely generated module over an artin algebra A is given, relative to an additive subbifunctor F of Ext1 (-, -) and an F-cotilting module T.
