$f$ is in the image of the function$j:R^* \to (R \to R)$ from the free monoid$R^*$ on $R$, i.e. the set of lists of elements in $R$, to the function algebra$R \to R$, such that

$j(\epsilon) = 0$, where $0$ is the zero function.

for all $a \in R^*$ and $b \in R^*$, $j(a b) = j(a) + j(b) \cdot (-)^{\mathrm{len}(a)}$, where $(-)^n$ is the $n$-th power function for $n \in \mathbb{N}$

for all $r \in R$, $j(r) = c_r$, where $c_r$ is the constant function whose value is always $r$.

$f$ is in the image of the canonical ring homomorphism$i:R[x] \to (R \to R)$ from the polynomial ring in one indeterminant $R[x]$ to the function algebra$R \to R$, which takes constant polynomials in $R[x]$ to constant functions in $R \to R$ and the indeterminant $x$ in $R[x]$ to the identity function $\mathrm{id}_R$ in $R \to R$

With scalar coefficients

For a commutative ring$R$, a polynomial function is a function$f:R \to R$ with a natural number$n \in \mathbb{N}$ and a function $a:[0, n] \to R$ from the set of natural numbers less than or equal to $n$ to $R$, such that for all $x \in R$,

$f(x) = \sum_{i:[0, n]} a(i) \cdot x^i$

where $x^i$ is the $i$-th power function for multiplication.

In non-commutative algebras

For a commutative ring$R$ and a $R$-non-commutative algebra$A$, a $R$-polynomial function is a function$f:A \to A$ with a natural number$n \in \mathbb{N}$ and a function $a:[0, n] \to R$ from the set of natural numbers less than or equal to $n$ to $R$, such that for all $x \in A$,

$f(x) = \sum_{i:[0, n]} a(i) x^i$

where $x^i$ is the $i$-th power function for the (non-commutative) multiplication.