Proof that $f\in R[X]$ with $f(u)=u^{-1}$ exists for commutative ring $R$

Question

Let $R$ be a commutative ring and $U\subset R^*$ a finite subset. How do I prove that there exists an $f\in R[X]$ with $f(u)=u^{-1}$ for all $u\in U$?

Answer 1

Let $u_1,\ldots,u_n$ be the units. Take the polynomial

$$g(X) = -\frac{\prod_{i=1}^n (X - u_i)}{\prod_{i=1}^n u_i} (-1)^n + 1.$$

It has vanishing constant coefficient, so is $g(X) = X f(X)$. Now $g(u_i) = 1$ for all $i$, so $u_i f(u_i) = 1$ and $f(X)$ is the sought for polynomial.