平坦模(flat module)是一类重要的模，投射模一定是平坦模，反之不一定成立。环A上每个左A平坦模是投射模的充分必要条件是，环A是左完全环。
再者，引进了分次半平坦模的概念，并有如下主要结果：环K是分次弱正则的当且仅当所有右K-模是分次半平坦的。
Next, the concept of graded semiflat module is introduced and proved that K is a group-graded weakly regular ring if and only if all right K-module is graded semiflat module.
解决的主要问题是把模的平坦分解推广为FP-平坦分解，利用维数从另一个角度来描述FP-平坦模的一些重要性质。
The main problems solved are that the flat decompositions are generalized to FP-flat decompositions, and some important properties of FP-flat modulus are described through dimensions in another way.
本文引入直投射覆盖的概念，证明了环R为左完全环当且仅当每一个左R-模（平坦左R-模）具有直投射覆盖；
This peper introduces the concept of direct-projective covers, and prove that a ring R is left perfect if and only if every left R-module(flat left R-module) has a direct-projective cover;
应用推荐