Each $c \in R^*$ divides each polynomial of $R[X]$

Question

We consider that $R$ is a commutative ring with $1_R$.

Each $c \in R^*$(if we see it as a constant polynomial), divides each polynomial of $R[X]$.

($c \in R^*$ means that $c$ is invertible.)

I haven't undersotod it..Could you explain it to me?

Does it mean that if we have a polynomial $p(X) \in R$,then $\frac{p}{c} \in \mathbb{Z}$ ? If yes, why is it like that??

Answer 1

If $c\in R^*$ then polynomial $f=a_{0}+\cdots+a_{n}X^{n}$ can be written as $cg$ where $g=c^{-1}a_{0}+\cdots+c^{-1}a_{n}X^{n}$.

Answer 2

If "reversible" mean "invertible," then this is true because invertible elements divide everything. (This is true in any ring with units and doesn't depend on the polynomial ring.)

If $u$ is invertible, then $a=(au^{-1})u$, and this says that $u|a$.

Answer 3

Hint $\ $ An invertible element remains invertible in every extension ring (having the same $\,\color{#c00}1)$. Therefore $\,c c' = \color{#c00}1\,$ for $\,c'\in R\,$ yields $\, f = \color{#C00}1\cdot f = (cc')f = c(c'f),\,$ so $\,c\mid f\,$ in $\,R[x].$