用直投射模刻划完全环和半完全环

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；当且仅当（有限生成）拟投射左Ｒ－模的直极限为直投射模。本文还证明了环Ｒ为半完全环当且仅当每一个有限生成（由２个元素生成的）左Ｒ－模具有直投射覆盖；当且仅当对所有自然数ｎ（存在自然数ｎ＞１）使得每一个循环左Ｒ＿ｎ－模具有直投射覆盖，这里Ｒ＿ｎ为环Ｒ上的ｎ阶全阵环。

Characterizations of Perfect Rings and Semiperfect Rings Using Direct-projective Modules

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;if and only if a direct limit of(finitely generated) quasi-projective R-modules is direct-projective.It is also shown that a ring R is semiperfect if and only if every finitely generated(2-generated) left R-moedule had a direct-projective cover,if and only if for all natural numbers n(there exists a natural number n>1) such that every cyclic left Rn-module has a direct-projective cover,where Rn denotes the ring of all n by n matrices over R.