Problem: Let $R$ be a domain. Prove that if a polynomial in $R[x]$ is a unit, then it is a nonzero constant (the converse is true if $R$ is a field).
My attempt: We proof that by contradiction. Suppose $u \in R$ be a unit of $R$ but $u$ is not a nonzero constant, then we have $\deg (u) > 0$.
$\forall f \in R$, $\deg (uf) \leq \deg (u) + \deg (f)$.
On the other hand, $u$ is a unit of $R$, so $\deg (uf) = \deg (f)$.
Associate with the inequality above, we see that the equality hold if and only if $\deg (u) = 0$. So we can conclude that $u$ is a nonzero constant.
Is my proof correct??? Thanks all!!!