Generalized D3(C3?C4)-Modules

As a proper generalization of D3-modules,the concept of generalized D3-modules(briefly G-D3 modules)is introduced in Chapter 3.A module M is said to be a G-D3 module if M₁is a direct summand of M,M₂is a direct summand of M,and M=M₁+M₂,then M₁?M₂is isomorphic to a direct summand of M.Basic properties of G-D3 modules are investigated.It is shown that a right hereditary ring R is a semisimple ring if and only if every right R-module is a G-D3 module if and only if every factor module of an injective right R-module is a G-D3 module;a right hereditary ring R is a rightV-ring if and only if every finitely cogenerated right R-module is a G-D3 module if and only if every finitely copresented right R-module is a G-D3 module.In Chapter 4 and 5,the concepts of generalized C3-modules(briefly G-C3 modules)and generalizedC4-modules(briefly G-C4 modules),the generalization ofC3-modules andC4-modules,are introduced.Basic properties of these modules are studied and the relationships between them and some related modules are discussed.