Definition:Polynomial over Ring

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

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

Thus:
 * Let $P \in R[X]$ be a polynomial

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

Also see

 * Polynomial is Linear Combination of Monomials
 * Definition:Polynomial Ring
 * Definition:Polynomial Function
 * Definition:Polynomial Equation
 * Definition:Polynomial Expression

Terminology

 * Definition:Coefficient of Polynomial
 * Definition:Degree of Polynomial
 * Definition:Null Polynomial