Find idempotents in a set of mappings 
Find idempotent elements of $S$ the set of maps $f:X \rightarrow R$ where $X$ is a given set and $R$ is a given ring with the operations defined by $(f+g)(r)=f(r)+g(r)$ and $(fg)(r)=f(r)g(r)$ where the sum and the product in these expressions are taken in $R$.

$f\in S$ is idempotent if $f^2=f$. Then for any element $x\in X$, we have that $f^2(x)=(f\cdot f)(x)=f(x)f(x)=[f(x)]^2$.
Hence the problem is when the equality $[f(x)]^2=f(x)$ holds. Since $R$ is a ring, we can add the opposite in both sides and we have that $[f(x)]^2-f(x)=0$ where $0$ is the neutral element in $R$. 
But, the only nilpotent elements are $0$ and $1$ that are the neutral elements for the addition and multiplication on $R$ respectively? Or there is any more?
Is there a way to proove that if $R$ does not have idempotents, the ring $S$ does not have idempotents either? And about nilpotents?
 A: Let $E$ be the set of idempotent elements of $R$.

Then $S=\{f:X\to R\mid f(x)\in E,\forall x\in X\}$.

In other words, $S$ is the set of functions from $X$ to $R$ whose range is a subset of $E$.

As noted in the comments, $R$ always has $0$ as an idempotent, and if $R$ has a multiplicative identity $1$ with $1\ne 0$, then $1$ is another idempotent of $R$.

You mentioned "nilpotent elements" but this problem is about idempotent elements, not nilpotent elements. Note also that $1$ is idempotent but not nilpotent (provided $1\ne 0$).

I hope that clears up a few things, but feel free to ask for further clarification.

Let's look at a few examples . . .

Example $(1)$:$\;$Let $R=\mathbb{Z}$ and let $X=\{a,b,c\}$.

Then $E=\{0,1\}$ and $S$ is the set of functions from $X$ to $R$ such that
$$
\begin{cases}
f(a)\in\{0,1\}\\[4pt]
f(b)\in\{0,1\}\\[4pt]
f(c)\in\{0,1\}\\
\end{cases}
$$
so $|S|=2^3=8$.

Example $(2)$:$\;$Let $R=\mathbb{Z_6}=\{0,1,2,3,4,5\}$ and let $X=\{a,b\}$.

Then $E=\{0,1,3,4\}$ and $S$ is the set of functions from $X$ to $R$ such that
$$
\begin{cases}
f(a)\in \{0,1,3,4\}\\[4pt]
f(b)\in \{0,1,3,4\}\\[4pt]
\end{cases}
$$
so $|S|=4^2=16$.

Example $(3)$:$\;$Let $R$ be the ideal $(2)=\{0,2,4\}$ of $\mathbb{Z_6}=\{0,1,2,3,4,5\}$ and let $X=\{a,b\}$.

Then $E=\{0,4\}$ and $S$ is the set of functions from $X$ to $R$ such that
$$
\begin{cases}
f(a)\in \{0,4\}\\[4pt]
f(b)\in \{0,4\}\\[4pt]
\end{cases}
$$
so $|S|=2^2=4$. Note that for this example, $R$ does not have a multiplicative identity.
