GENERALIZATIONS OF C3 MODULES AND C4 MODULES

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 boolean AND K = 0 and K is an element of A, then N circle plus K is also a direct summand of M; M is called an A-C4 module, if whenever M = A circle plus B where A and B are submodules of M and A is an element of 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.