| Abstrakt: |
Let 풰 and 풩 be two families of R-modules, V a submodule of a direct sum of some elements in 풰, and X a submodule of a direct sum of some elements in 풩. An R-module 풩 is 풰푉-generated if there is an epimorphism from V to 풩. in the other hand, the family 풩 is an X-sub-linearly independent to an R-module M if there is a monomorphism from X to M. The concept of 풰푉-generated module and X-sub linearly independent are used to define 풰-basis and 풰-free module. A projective풰 module is a generalization of a projective module which is the direct summand of a 풰-free module. In this paper, we construct the example of the projective풰 module for some family 풰 of R-modules. Moreover, we investigate the properties of the projective풰 module. Based on research, we have every 풰-free module is a projective풰 module, and 푅-module 0 is a strictly projective풰 module. Furthermore, if 푃1, 푃2, ..., 푃푛 are projective풰 modules, then ⊕ i = 1 n P i is a projective풰 module. If 푃 is semisimple module, then every submodule of 푃 is a projective풰 module, and every direct summad of a projective풰 module is a 풰V-generated module. [ABSTRACT FROM AUTHOR] |
