ON GENERALIZED EXTENDING MODULES

and K. Oshiro have introduced, in a diagram of modules and homomorphisms, the concept of generalized M-injective modules. S. Mohamed, and B. Mueller have given a different characterization, based on an exchange property, of the generalized M-injective modules. Here we introduce the concept of M-jective modules, which is a generalization of Mohamed and Mueller concept for the generalized M-injectivity. The concept of M-jective modules is used here to solve the problem of finding a necessary and sufficient condition for a direct sum of extending modules to be extending. In fact, we show that relative jectivity is necessary and sufficient for a direct sum of two extending modules to be extending. We also introduced the concept of generalized extending modules, and give some properties of such modules in analougy with the known properties for extending modules. 1. Introduction. In [2], H. Hanada, J. Kado, and K. Oshiro have introduced the concept of general-ized M-injective modules, which is a generalization to the concept ofM-injectiv