Let $X$ be a (not reduced) connected curve (over a base-field $k$), and $X_{red}$ be the corresponding reduced curve (in the sense of Liu, 2.4.2).
It if known (Liu, 3.3.21), that $\mathcal{O}_{X_{red}}(X_{red})$ is a finite field extension of $k$.
Furthermore (Liu, 2.4.2), $\mathcal{O}_X(X) / \sqrt{0_{\mathcal{O}_X(X)}}$ injectively embeds in the field $\mathcal{O}_{X_{red}}(X_{red})$.
Question: Are there examples where this is a proper subset?
I have tried to construct a curve $X$ such that $\mathcal{O}_X(X) \cong \mathbb{Q}[t] / (t^2)$ and $\mathcal{O}_{X_{red}}(X_{red}) \cong \mathbb{Q}[\sqrt{2}]$. My idea was to start with a curve which has $\mathbb{Q}[\sqrt{2}]$ as global sections, i.e. the gluing of $R_0 = \mathbb{Q}[x,y] / (y^2-2)$ and $R_\infty = \mathbb{Q}[1/x,y] / (y^2-2)$ along the localisation $(R_0)_x \cong (R_\infty)_{1/x}$, and then square the relations, i.e. $S_0 = \mathbb{Q}[x,y] / ((y^2-2)^2)$ and $S_\infty = \mathbb{Q}[1/x,y] / ((y^2-2)^2)$. But then the global sections are $\mathbb{Q}[y] / ((y^2-2)^2)$, and that modulo the nilradical is $\mathbb{Q}[y] / (y^2-2)$, which is to big.