### Finitely Generated Z-modules

Definition: Let R be a commutative ring with identity, let M be an R-module, and let B be a nonempty subset of M. Then the set RB is defined as RB is a submodule. If B is a finite set, say , we write for RB, and say that RB is a finitely generated R-module. In particular, if for some , we say that M is finitely generate