R-module

Let R be an associative ring with identity and M a right R-module .

设R是有单位元的结合环，M是右R－模。

( vi ) every finitely presented torsionless right R-module is reflexive ;

（Vi）每个有限表现半自反右R－模是自反的；

R is left IF ring iff every finitely generated left R-module is embeddable in a flat module .

R为左IF环当且仅当任一有限生成左R&模均可嵌入平坦模；

R-module M is regular if and only if ( 1 ) M is locally Projective ;

R-模M是正则的当且仅当（1）M是局部投影的；

Finitely generated free R-module

有限生成自由R－模

Finitely generated graded R-module

有限生成分次R－模

Finitely generated projective R-module

有限生成射影R－模

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 ;

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 paper proves that when T - is a Spectral Topology , R as a R-module is a r - finite dimensional module , if and only if RT - is a Semi-simple Artin Ring .

证明了在τ是谱拓扑的情形下，R作为R模是τ有限维的当且仅当Rτ是半单Artin环。

R / Soc ( R_R ) is a right V-ring if and only if every proper essential right ideal E of R is an intersection of maximal right ideals , if and only if there exists a singular semisimple right R-module which is quasi-dual .

设R为环，我们考虑下面两个条件。2每一个R的本质真右理想是极大右理想的交；3存在一个奇异半单右R-模是quasi-dual的。

A ring R is said to be a FCGP-ring if each FCG-projective R-module is projective , some equivalent conditions of FCG-projective modules are given , left V-rings and semisimple rings are characterized by FCG-projective modules , characters and equivalent conditions of FCGP-rings are discussed .

环R称为FCGP环，如果任一FCG投射R模都为投射模.给出了FCG投射模的等价条件，并用FCG投射模刻画了左V环和半单环。