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A Finitely Generated Module over a Valuation Domain

This article discusses about some properties which are equivalent between a finitely generated module over PID and a finitely generated module over a valuation domain. This can be done by considering a finitely generated module over a DVR. Although in general a PID is not a valuation domain or vice versa, these equivalence of some properties will be valid. It is because a DVR is a PID and a valuation domain at the same time. Those the equivalent properties in a finitely generated module over DVR are related with the decomposition of the module and the height of an element in that module.