Ring generated by binomial polynomials

Definition

Let $R$ be a commutative unital ring of characteristic zero. Let $K$ be the quotient ring of $R$ by the multiplicative subset of nonzero integers. Then, the ring generated by binomial polynomials over $R$ is the subring of $K[x]$ comprising all $R$-linear combinations of the polynomials:

$(\genfrac{}{}{0ex}{}{x}{r})=\frac{x(x-1)(x-2)\dots (x-r+1)}{r!}$.

where $r\ge 0$ (for $r=0$, this is the constant polynomial $1$).

Equivalently, it is the tensor product with $R$ of the ring generated by binomial polynomials over the rational integers, i.e., the ring generated by binomial polynomials over $\mathbb{Z}$.

Equivalently, it is the ring $\mathrm{I}\mathrm{n}\mathrm{t}(\mathbb{Z},R)$: the ring of all polynomials $f\in K[x]$ such that $f(\mathbb{Z})\subseteq R$.

Facts

• Ring of integer-valued polynomials is contained in ring generated by binomial polynomials
• Ring of integer-valued polynomials over rational integers equals ring generated by binomial polynomials