(N,m)-strong F-Gorenstein Projective Modules And Weakly F-Gorenstein Algebra

For projective,inject.ive and flat modules in general homological alge-bras,there are corresponding definitions such as Gorenstein projective mod-ules,Gorenstein injective modules and Gorenstein flat modules.And then there are strong Gorenstein projective modules,strong Gorenstein injective modules and strong Gorenstein flat,modules.In this paper,suppose F is an additive sub-bifunetor of ExtA1(-,-),F has enough projective and injec-tive modules.Based on the study of Gorenstein homological algebras,we further study(n.m)-strong F-Gorenstein projective modules and weakly F-Gorenstein homological algebras,and give the properties related to(n,m)-strong F-Gorenstein projective modules and equivalent characterizations of weakly F-Gorenst.ein homological algebras.This paper is divided into three parts:The fir-st part introduces some basic knowledge about Gorenstein homological algebras and F-relative Goren-stein homological algebras.In the second part.,we study the relations between n-strong F-Gorenstein projective modules and m-strong F-Gorenst.ein projec-tive modules when n is not,equivalent to m.Xext the definition of(n,m)-strong F-Gorenstein projective modules is given.It is proved that(n,m)?strong F-Gorenstein projective modules is closed under direct sum,but,the direct summand is rnot closed in gerneral.The relationship between(n,m)?strong F-Gorenstein projective module and its syzygy are also considered.In the last,we give a special condition that the direct summand of(n,m)-strong F-Gorenstein projective modules is closed under the direct,summand.In the third part,we define F-semi-Gorenstein projective module,and use it to de-fine weakly F-Gorenstein algebra We give some equivalent characterizations of weakly F-Gorenstein algebra in the end.