**Problem.** Let be an infinite ring with 1 and let be the set of units of Prove that if is finite, then is a division ring.

**Solution**. Suppose, to the contrary, that there exists some First note that if is a left or right ideal of then is finite because otherwise and so Therefore and cannot both be infinite. Suppose that is finite and let Then is a left ideal of and because Hence is finite. Now the -module isomorphism implies that is finite. Contradiction!

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