Characterizations Of δ-Perfect (Semiperfect,Semiregular,Semilocal) Rings

The concept of D4-δ-covers is introduced in chapter 3,a pair(F,g)is called a D4-δ-cover of a module M,if F is a D4-module,and g an epimorphism of F onto M with Ker(g)<<δ F.The basic properties of D4-δ-covers and relationships between D4-δ-covers with projective δ-covers are studied.Let g:F→M be an R-epimorp-hism,where M is an R-module and F is a projective R-module,then M has a proj-ective 8-cover if and only if F⊕M has a D4-δ-cover.The connections between D4-δ-covers and δ-lifting modules are investigated.If M is a projective module and N≤M,then M is δ-lifting if and only if M ⊕(M/N)has a D4-δ-cover.If M is a projective module,then every factor module of M ⊕ M has a D4-δ-cover if and only if M is δ-lifting.δ-perfect(semiperfect,semiregular)rings are characterized by using D4-δ-covers.It is shown that a ring R is δ-perfect if and only if every right R-module has a D4-δ-cover;R is δ-semiperfect if and only if every generated R-module has a D4-δ-cover;R is δ-semiregular if and only if every finitely presented R-module has a D4-δ-coverThe concepts of GSδ-module and GASδ-module are introduced on the basis of GS-modules and GAS-modules in chapter 4.A module M is said to be a GSδ-module if for any submodule N ≤M,there exists L ≤M such that M-N+L and N∩L≤δ(L).M is called a GASδ-module if M=N+L implies that N has a GSδ-supplement H and H ≤ L.We prove that if M=M1+M2,M1,M2 are GSδmodules,then M is a GSδ-module.If M is a GASδ-module and K a direct summand of M,then K is a GASδ-module.If M=U1+U2,U1,U2 have GASδ-supplements in M,then U1∩U2 has a GASδ-supplement in M.M is Artinian if and only if M is a GASδ-module and satisfies DCC on generalized δ-supplemented submodules and on δ-small submodules.If M is a finitely generated GASδ-module,then M is Artinian if and only if M satisfies DCC on δ-small submodulesIn chapter 5,the concepts of WGSδ-modules is introduced on the basis of WGS-module.A module M is said to be a WGSδ-module for any submodule N ≤M,there exists L ≤M such that M=N+L and N∩L≤δ(M).It is proven that if M is a WGSδ-module,then every δ-supplemented submodule of M is a WGSδ-module.If f:N→M is a δ-cover of WGSδ-module M,then N is a WGSδ-module.Every factor module of a WGSδ-module is a WGSδ-module.The connections between WGSδ-modules and 8-weakly supplemented modules are studied.If M is finitely generated,then M is a WGSδ-module if and only if M is a δ-weakly supplemented module.We prove that if M-M1+M2,M1,M2 are WGSδ-modules,then M is a WGSδ-module.M is a WGSδ-module if and only if M/δ(M)is semisimple if and only if there is a decomposition M=M1⊕M2 such that M1 is semisimple,δ(M)≤e M2 and M2/δ(M)is semisimple.It is shown that a ring R is δ-semilocal if and only if every right R-module is a WGSδ-module.