Divided polynomial ring

Definition

Let $R$ be a commutative unital ring. The divided polynomial ring in one variable with indeterminate $x$ over $R$, also called the free divided power algebra in one variable, is defined as the ring obtained by adjoining formal symbols $x^{(n)}$ for all natural numbers $n$ to $R$, subject to the following relations for all natural numbers $n$ and all $i$ with $0<i<n$:

$x^{(i)}x^{(n-i)}={\binom {n}{i}}x^{(n)}$

We can additionally set $x^{(0)}=1$ (so that the above becomes true with $0\leq i\leq n$) and we denote $x^{(1)}$ by $x$.

Particular cases

• In the case that $R$ is a $\mathbb {Q}$-algebra, the divided polynomial ring is the same as $R[x]$, and the element $x^{(n)}$ is identified with $x^{n}/n!$.
• In case the characteristic of $R$ is zero, we can realize the divided polynomial ring as an intermediate subring between $R[x]$ and $L[x]$, where $L$ is the localization of $R$ at the multiplicatively closed subset of nonzero integers. Explicitly, $x^{(n)}=x^{n}/n!$, which makes sense inside $L[x]$.

Related notions

• Ring generated by binomial polynomials
• Ring of integer-valued polynomials