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This question is from my module theory assignment and I am struck on this particular problem.

Prove that if D is a ring with identity such that every D-module is free, then D is a divison ring.

Attempt: To prove that D is Divison Ring it is sufficient to show that D has no proper ideals. Let on the contrary D has a proper ideal I. Now , I have to somehow prove that some D-module is not free. But I am not able to get any ideas.

Can you please help?

  • If you know that $D$ has a proper ideal, what $D$-modules come to your mind? – svelaz Oct 14 '21 at 05:03
  • Please search before asking. This was available in the Related questions to the right, and so it should have appeared when you were typing your original question in. – rschwieb Oct 14 '21 at 12:18

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