Im given the following task: Let X be a free Z-Module with a basis containing of one element. Show that every submodul of X is free. Obviously X is isomorphic to Z. So if I assume X=Z the proof is fairly trivial. However, why does this then also apply for an arbitrary X? Isomorphisms are known for keeping the structure, but why exactly is this the case here? I don't immediately see it. Thank you in advance for answers!
2026-03-25 15:10:42.1774451442
Isomorphism and free submodule
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