Sheaf of rings on a discrete set. I was reading through some notes for an exam and one exericse ask

Sheaf of rings on a discrete set.
I was reading through some notes for an exam and one exericse asks me to prove the following
There is a unique sheaf of rings making a topological set X with discrete topology a ringed space.
I tried doing it but I feel I'm missing something, using the definition of presheaf and than of sheaf doesn't seem to bring me any result. How can I solve such a problem? I leave you my definitions of presheaf and sheaf.
A presheaf $\mathfrak{F}$ (of rings) on a topological space X consists of the data:
for every open set $U\subset <!--\; \subset \; -->X$ a ring $\mathfrak{F}(U)$ (think of this as the ring of functions on U),
for every inclusion $U\subset <!--\; \subset \; -->V$ of open sets in X a ring homomorphism ${\mathrm{\rho <!--\; \rho \; -->}}_{V,U}:\mathfrak{F}(V)\to <!--\; \to \; -->\mathfrak{F}(U)$ called the restriction map (think of this as the usual restriction of functions to a subset), such that
$\mathfrak{F}(\mathrm{\varnothing <!--\; \varnothing \; -->})=0$
${\mathrm{\rho <!--\; \rho \; -->}}_{U,U}$ is the identity map of $\mathfrak{F}(U)$ for all U,
for any inclusion $U\subset <!--\; \subset \; -->V\subset <!--\; \subset \; -->W$ of open sets in X we have ${\mathrm{\rho <!--\; \rho \; -->}}_{V,U}\circ <!--\; \circ \; -->{\mathrm{\rho <!--\; \rho \; -->}}_{W,V}={\mathrm{\rho <!--\; \rho \; -->}}_{W,U}$
The elements of $\mathfrak{F}(U)$ are usually called the sections of F over U, and the restriction maps ρV,U are written as $\mathrm{\phi <!--\; \phi \; -->}\to <!--\; \to \; -->\mathrm{\phi <!--\; \phi \; -->}|U$
A presheaf $\mathfrak{F}$ is called a sheaf of rings if it satisfies the following gluing property:
if $\subset <!--\; \subset \; -->X$ is an open set, $\{U_i:i\in <!--\; \in \; -->I\}$ an arbitrary open cover of U and ${\mathrm{\phi <!--\; \phi \; -->}}_i\in <!--\; \in \; -->\mathfrak{F}(U_i)$ sections for all i such that ${\mathrm{\phi <!--\; \phi \; -->}}_i{|}_{U_i\cap <!--\; \cap \; -->U_j}={\mathrm{\phi <!--\; \phi \; -->}}_j{|}_{U_i\cap <!--\; \cap \; -->U_j}$ for all i,$i,j\in <!--\; \in \; -->I$, then there is a unique $\mathrm{\phi <!--\; \phi \; -->}\in <!--\; \in \; -->\mathfrak{F}(U)$ such that $\mathrm{\phi <!--\; \phi \; -->}{|}_{U_i}={\mathrm{\phi <!--\; \phi \; -->}}_i$ for all i.
EDIT: This is what is given as the definition of a K-ringed space:
A ringed spaces equipped with a sheaf of rings such that the elements of ${\mathfrak{O}}_X(U)$ are actual functions from U to a fixed ring K;
EDIT: It turns out that the actual definition is
A ringed spaces equipped with a sheaf of rings such that the elements of ${\mathfrak{O}}_X(U)$ are actual functions from U to a fixed ring K and ${\mathfrak{O}}_X(U)$ is not only a subring of the ring of functions from U→K but a sub−K−algebra of it;
What does this change?