Generalizations of C3 Modules and C4 Modules

Abstract

"Let A be a class of right R-modules that is closed under isomorphisms, and let M be a right R-module. Then M is called A -C3 if, whenever N and K are direct summands of M with N ∩K = 0 and K ∈ A , then N ⊕K is also a direct summand of M; M is called an A -C4 module, if whenever M = A⊕B where A and B are submodules of M and A ∈ A , then every monomorphism f : A → B splits. Some characterizations and properties of these classes of modules are investigated. As applications, some new characterizations of semisimple artinian rings, right V-rings, quasi-Frobenius rings and von Neumann regular rings are given."