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 $$0 \le i \le n$$) 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