Quotient-transitivity and cyclic submodule-transitivity for $p$-adic modules
Two new notions of transitivity, which we have named quotient-transitivity and transitivity with respect to cyclic submodules for $p$-adic modules, are introduced. Unlike the classical notions that derive from Abelian group theory, this approach is based on isomorphism of quotients and makes no use of height sequences. The two new notions lead to a sufficiently large class of interesting modules. Our principal result is that finitely generated $p$-adic modules are both quotient-transitive and transitive with respect to cyclic submodules.