Abstract Algebra II · Modules and finitely generated modules over a PID
When finitely generated torsion means finite
Problem
Prove that every finitely generated torsion $\mathbb Z$-module is finite. Then explain why the same wording is false for modules over an arbitrary PID with infinite residue fields.
Hint
A finitely generated torsion $\mathbb Z$-module is a finite direct sum of groups $\mathbb Z/d_i\mathbb Z$.
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