Let $R$ be a ring and $A$ be a non-empty set. The ring of functions from $A$ to $R$ is the ring $\fun(A, R)$ 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
\[\underline{0} : a \mapsto 0, \quad \underline{1} : a \mapsto 1.\]