Let R be a Dedekind domain. A be a finitely generated R-module. Then A = A 1 ⊕ A 2 , for some torsion module A 1 and torsion-free module A 2. Proof that A1 has finite composition length. I can see why A1 has ACC, but i can not proof A1 has DCC. This is from: Module Theory, Extending Modules and Generalizations. Theorem 4.12.