LetRbe a commutative ring and letFbe a filter of ideals ofR. We give three characterizations of the rings with the property that all finitely generated F pretorsionRmodules decompose into a direct sum of cyclicRsubmodules. One of these characterizations involves decomposing the filterFinto a product of subfilters. This filter decomposition is shown to be unique, and is used to characterize several other ring properties relative to the filter of ideals.