Definition:Polynomial over Ring/One Variable

Definition
Let $R$ be a commutative ring with unity.

A polynomial over $R$ in one variable is an element of a polynomial ring in one variable over $R$.

Thus:
 * Let $P \in R \left[{X}\right]$ be a polynomial

is a short way of saying:
 * Let $R \left[{X}\right]$ be a polynomial ring in one variable over $R$, call its variable $X$, and let $P$ be an element of this ring.

Also see

 * Definition:Polynomial over Ring in Multiple Variables