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 $0<i<n$:

$x^{(i)}{x}^{(n-i)}=(\genfrac{}{}{0ex}{}{n}{i})x^{(n)}$

We can additionally set $x^{(0)}=1$ (so that the above becomes true with $0\le i\le n$) 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