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.

Structural definition

For a commutative ring$R$, let $R[\mathbb{1}]$ be thefree commutative $R$-algebra on the singleton$\mathbb{1}$ with element $0 \in \mathbb{1}$, with canonical function $x:\mathbb{1} \to R[\mathbb{1}]$, and let $R \to R$ be the function algebra of $R$, with identity function$id_R:R \to R$. Since both objects are commutative $R$-algebras there are canonical commutative ring homomorphisms$\alpha:R \to R[x]$ and $\beta:R \to (R \to R)$, and there exists a commutative ring homomorphism $i:R[\mathbb{1}] \to (R \to R)$ such that $i \circ \alpha = \beta$ and $i(x(0)) = id_R$. A polynomial function is a function $f:R \to R$ in the image of $i:R[\mathbb{1}] \to (R \to R)$.

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.