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