Infinite Abelian Groups, Numéro 2

abelian groups algebraically compact automorphism Baer bounded order cardinal number characteristic submodules compact abelian group complete direct sum complete discrete valuation countable cyclic direct summand cyclic groups cyclic modules decomposition define direct summand discrete valuation ring divisible group divisible subgroup elements of infinite elements of order exercise finite group finite number fully invariant submodules fully transitive G and H group G group of bounded h(px Hence homomorphic image hypothesis indecomposable infinite height isomorphic to Z(p Let G linear transformation mapping multiple p-adic integers polynomial primary module prime principal ideal ring proof of Theorem pure subgroup R-module rational numbers reduced primary group ring of endomorphisms satisfying Show subgroup of G sum of cyclic summand of G Suppose T₁ Theorem 13 Theorem 28 theory torsion group torsion subgroup torsion-free modules transfinite U-sequence Ulm invariants Ulm sequence Ulm's theorem unique vector space write Z₂ Zorn's lemma