Completions of commutative rings and modules

In this work we continue studying the notion of completion ofR-modules, over a commutative ringR, relative to a torsion theoryϑ. We develop some techniques relative to localization at prime ideals and give structural results on the completion of finitely generatedR-modules, describing it as the product of classical completions on local noetherian rings. The authors acknowledge partial support from the D.G.I.C.Y T.