Fully inert submodules of torsion-free modules over the ring of \(p\)-adic integers (Q2879355)

Let \(J_{p}\) denote the ring of \(p\)-adic integers. In analogy to the case of subgroups of abelian groups, a submodule \(H\) of a \(J_{p}\)-module \(G\) is said to be fully inert if, for any endomorphism \(\phi\) of \(G\), \((H+ \phi H)/H\) is finite as \(J_{p}\)-module. Two submodules \(A\) and \(B\) of a \(J_{p}\)-module \(G\) are said to be commensurable if \((A+B)/A\) and \((A+B)/B\) are both finite. The fully inert subgroups of a free \(J_{p}\)-module of infinite rank are characterized. A fully inert submodule of a free or a complete torsion-free \(J_{p}\)-module is shown to be commensurable with a fully invariant submodule, but examples show that the requirement of completeness cannot be dropped in the case of torsion-free \(J_{p}\)-modules.