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Let U be a family of R-modules and V be a submodule of a direct sum of some elements in U. The aim of this paper is to generalize basis and free modules. We use the concept of U v-generated module and X-sublinearly independent to provide the definition of U-basis and U-free module. We construct a U -basis of an R-module M as a pair (X, V), which a family U is X-sub-linearly independent to M and M is a Uv-generated module. Furthermore, we define Ubasis of M as a U -basis which has the maximaldoi:10.17656/jzs.10287 fatcat:zgv6s4yc5bbyrg6vhcoc7yv7ma