2.3) A scheme $(X,\mathcal{O}_X)$ is reduced if for every open set $U\subset X$, the ring $\mathcal{O}_(U)$ has no nilpotent element.
b) Let $(X,\mathcal{O}_X)$ be a scheme. Let $(\mathcal{O}_X)_{red}$ be the sheaf associated to the prescheaf $U\rightarrow \mathcal{O}_X(U)_{red}$, where for any ring $A$, we denote by $A_{red}$ the quotient of $A$ by its ideal of nilpotent elements. Show that $(X,(\mathcal{O}_X)_{red})$ is a scheme.
P.S. I have been able to show that it is a locally ringed space. How to proceed further.
Here's a possible hint--show that this is the quotient sheaf associated to the ideal sheaf $U\mapsto \text{nil}(\mathcal{O}_X(U))$. Show that this sheaf is quasicoherent.