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 generated, and that is a generating set for M.
Exercise: Show that the and are finitely generated, by giving a finite generating set for each. Where Z stands for the integers.
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Finitely Generated Z-modules are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.