Let $R$ be a ring and $A$ be a non-empty set. The ring of functions from $A$ to $R$, $\fun(A,R)$, is the ring with addition and multiplication defined by
\[\begin{align*} f+g &: a\mapsto f(a)+g(a), \\ fg &: a\mapsto f(a)g(a) \end{align*}\]for all $f$, $g\in\fun(A,R)$. The neutral elements are the constant functions
\[\underline{0} : a\mapsto 0, \quad \underline{1} : a\mapsto 1.\]