The Stacks project

Example 15.23.17. The results above and below suggest reflexivity is related to the $(S_2)$ condition; here is an example to prevent too optimistic conjectures. Let $k$ be a field. Let $R$ be the $k$-subalgebra of $k[x, y]$ generated by $1, y, x^2, xy, x^3$. Then $R$ is not $(S_2)$. So $R$ as an $R$-module is an example of a reflexive $R$-module which is not $(S_2)$. Let $M = k[x, y]$ viewed as an $R$-module. Then $M$ is a reflexive $R$-module because

and $M$ is $(S_2)$ as an $R$-module (computations omitted). Thus $R$ is a Noetherian domain possessing a reflexive $(S_2)$ module but $R$ is not $(S_2)$ itself.