ON THE EXISTENCE OF FLAT COVERS IN R-gr

Recently, a proof of the existence of a flat cover of any module over an arbitrary associative ring with unit has been finally given (see 4-5 Bican, L., EL Bashir, R. and Enochs, E. All Modules Have Flat Covers, preprint Eklof and Trlifaj, J. How to Make Ext Vanish. J. London Math. Soc., to appear ). In this paper we prove the existence of flat covers in the category of graded modules over a graded ring. Some graded theoretical machinery is introduced to make the proof possible and new graded homological tools are developed.