Integral domain with a finitely generated non-zero injective module is a field

Question

Suppose that $R$ is a integral domain. Suppose that there exists a non-zero finitely generated injective module $M$. How can I prove that $R$ is field?

Answer 1

Let $Q$ be a finitely generated injective $R$-module. Suppose that $R$ is not a field and let $\mathfrak m$ be a maximal ideal of $R$. Since $\mathfrak m\ne 0$ there is $a\in\mathfrak m$, $a\ne 0$. Then we have $aQ=Q$ (injective modules are divisible), and therefore $\mathfrak mQ=Q$. Localizing we get $\mathfrak mQ_{\mathfrak m}=Q_{\mathfrak m}$, and by Nakayama we get $Q_{\mathfrak m}=0$. Since all the localizations of $Q$ are zero we conclude that $Q=0$.