If any two nonzero elements of an integral domain R are associates, prove that R is a field?

Question

Question from abstract algebra course...I have already shown that if a=bu for some unit u and b=av for some unit v, by definition of associates, then u and v are inverses...but I can't quite make the jump to proving that every element of R is a unit. Any help would be appreciated!

Answer 1

You have that any two non-zero elements are associates right? So for any non zero element $r\in R$, $r$ and $1$ are associates (since $1$ is non-zero). This means that $1=ur=ru$ for some unit $u\in R$ (since integral domains are commutative). But this is precisely the definition of $r$ being a unit. Namely you have that $u=r^{-1}$. Hence every non-zero element of your integral domain is invertible so your integral domain is in fact a field.