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This is not for homework, but I seem to be stuck a would like a hint please. The question asks

If every nonzero element of an integral domain $R$ is either a unit or irreducible, then $R$ is a field.

The question looks non-threatening, and I'm surely missing something obvious. I started by choosing some nonzero $r \in R$. If $r$ is a unit then there is nothing to prove. If $r$ is irreducible, however, I don't immediately see how to conclude that $r$ is invertible. Any hints would be greatly appreciated!

tylerc0816
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Hint: If $r$ is irreducible, then what can be said about $r^2$? What does this imply about $r$?

Cameron Buie
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  • OK, now I would like someone to explain this to me! Thanks in advance! – Robert Lewis Oct 30 '13 at 19:05
  • Well, it's certainly not true that $r^2$ is irreducible, since it can be written as $r\cdot r,$ and neither of $r$ or $r$ (silly, I know, but true) is a unit. Furthermore, since $r\ne 0$ and $R$ is an integral domain, then $r^2\ne 0,$ and so $r^2$ is a unit by hypothesis. Hence, there is some $u\in R$ such that $r^2\cdot u=1,$ but then $r\cdot u\in R,$ and $r\cdot(r\cdot u)=r^2\cdot u=1,$ and so $r$ is a unit, giving us the desired contradiction. – Cameron Buie Oct 30 '13 at 19:09
  • To Cameron Buie: "neither $r$ nor $r$ is a unit" . . . that is good! Thank you very much, Professor Buie! – Robert Lewis Oct 30 '13 at 19:20