Let △ be a multiplicatively closed set of finitely generated nonzero ideals of a ring R. Then the concept of a △ -reduction of anR-submoduleDof anR-moduleAis introduced and several basic properties of such reductions are established. Among these are that a minimal △ -reductionBofDexists and that every minimal basis ofBcan be extended to a minimal basis of allR-submodules betweenBandD, whenRis local andAis a finiteR-module. Then, as an application, △ -reductionsBof a submodule C with property (*) are introduced, characterized, and shown to be quite plentiful. Here, (*) means that (R ,M) is a local ring of altitude at least one, that △ = {Mn;n≥ 0} and that ifD⊆EareR-submodules betweenBandC, then every minimal basis ofDcan be extended to a minimal basis ofE.
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