Restricted Projective,Injective And Flat Complexes

Over an associative ring R,firstly,the notions of restricted injective modules are in-troduced by derived functors.We show that Gorenstein injective modules are restricted injective modules,but not vice versa.The classical restricted injective dimension is inves-tigated using resolutions by restricted injective modules,and some new methods are given to compute the restricted injective dimension.The relationship between restricted injec-tive modules and strongly torsion-free modules are investigated.Secondly,the notions of restricted projective,restricted injective and restricted flat complexes are introduced,and some characterizations and properties are given.It is shown in the paper that a complex C is restricted projective if and only if Cm is a restricted projective module for each m?Z,and each morphism from C to L is homotopic to 0 whenever L is a complex of finite injective dimension.If furthermore C is bounded right,then C is restricted projective if and only if Cm is a restricted projective module for each m?Z.In addition,if C is an exact complex of R-modules,then C is restricted projective if and only if Zm(C)is a restricted projective module for each m ? Z.Similarly,restricted injective and restricted flat versions of all these results are given.