Let $D$ be a commutative domain with multiplicative identity $1$ and assume that the additive group $D$ is finitely generated. Prove if there exists an integer $n > 1$ such that $f: D \rightarrow D$ defined by $x \mapsto nx$ is onto, then $D$ is a finite field.
I want to confirm if this problem makes sense. To me it doesn't, because if we take $D$ to be $\mathbb{R}$, then any $n$ makes $f$ onto, but $\mathbb{R}$ isn't a finite field.
By assumption the additive group is finitely generated, so it is a direct product of finitely many cyclic groups. If one of these cyclic groups is infinite then $x\mapsto nx$ is not onto for every $n>1$. Hence the ring is finite.
A finite domain is a field. QED