Abstract
Let 
be a family of R-modules and V be a submodule of a direct sum of some elements in The aim of this paper is to generalize basis and free modules. We use the concept of v -generated module and X-sublinearly independent to provide the definition of -basis and -free module. We construct a ̲-basis of an R-module M as a pair (X, V), which a family is X-sub-linearly independent to M and M is a v -generated module. Furthermore, we define -basis of M as a ̲-basis which has the maximal element on the first component and the minimal element on the second component of a pair (X, V). The results show that the first component of (X, V) in -basis is closed under submodules and intersections. Moreover, we prove that the second component of (X, V) in -basis is closed under direct sums. We also determine some -free modules related to a family which contains all Z-module Z modulo p power of n, where p prime and n ≥ 2.
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