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In this paper, we consider the question of the complete faithfulness of the$p$-free quotient module of the dual Selmer groups of elliptic curves definedover a noncommutative $p$-adic Lie extension. Our question will refine previousquestions on the complete faithfulness of dual Selmer groups. We also considerthe question of the triviality of the central torsion submodules of theseIwasawa modules and we see that this latter question is intimately related tothe former. We will also formulate and study analogous questions for the dualSelmer groups of Hida deformations. We then give positive answer to ourquestions, and establish "control theorem" results between the questions incertain cases.

On the complete faithfulness of the $p$-free quotient modules of dual Selmer groups