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Direct sum of cyclic groups
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abelian groups additive group arbitrary Artinian ring assume automorphism Baer basic subgroup belongs cardinal complete direct sum completely decomposable components corresponding coset countable define denote direct decomposition direct summand divisible group elements of G elements of infinite endomorphism ring exists factor group finite group finite number finite rank follows fully invariant subgroup G contains group G groups of rank hence Hint homomorphic image Horn implies indecomposable induces infinite cyclic groups infinite height infinite order isomorphic Kulikov left ideals Lemma Let G mapping matrix maximal independent set maximal torsion subgroup minimum condition mixed group Mult G nil group non-isomorphic non-zero obtain p-adic integers p-components p-group G prime prove pure subgroup quasicyclic groups reduced p-group satisfy solvable subgroup of G subset sum of cyclic summand of G Theorem torsion free groups torsion free rank torsion group Ulm factors Ulm sequence