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edit approved Dec 15, 2021 at 15:48
Jakeup
edit approved Dec 15, 2021 at 15:48
Jakeup
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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 $A_2.$ Prove that $A_1$ has finite composition length.

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. II can see why A1$A_1$ has ACC, but iI can not proof A1prove $A_1$ has DCC. 
This is from: Module Theory, Extending Modules and Generalizations. Theorem 4.12. Module Theory, Extending Modules, and Generalizations. Theorem 4.12.

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.

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 $A_2.$ Prove that $A_1$ has finite composition length.

I can see why $A_1$ has ACC, but I can not prove $A_1$ has DCC. 
This is from Module Theory, Extending Modules, and Generalizations. Theorem 4.12.

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Na Man
Na Man
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Torsion module Finite composition length

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.