Units of Ring of Polynomial Forms over Commutative Ring

Theorem
Let $\left({R, +, \circ}\right)$ be a commutative ring with unity whose zero is $0_R$ and whose unity is $1_R$.

Let $R \left[{X}\right]$ be the ring of polynomial forms in an indeterminate $X$ over $R$.

Let $P \left({X}\right) = a_0 + a_1 X + \cdots + a_n X^n \in R \left[{X}\right]$.

Then $P \left({X}\right)$ is a unit of $R \left[{X}\right]$ iff $a_0$ is a unit of $R$, and for $i = 1, \ldots, n$, $a_i$ is nilpotent in $R$.

Corollary
Let $R$ be a reduced ring.

Let $R \left[{X}\right]$ be the ring of polynomial forms in an indeterminate $X$ over $R$.

The group of units of $R \left[{X}\right]$ is precisely the group of elements of $R \left[{X}\right]$ of degree zero that are units of $R$.