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.

Multiple Variables
Let $I$ be a set.

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

Thus:
 * Let $P \in R[(X_i)_{i\in I}]$ be a polynomial

is a short way of saying:
 * Let $R[(X_i)_{i\in I}]$ be a polynomial ring in $I$ variables over $R$, call its variables $(X_i)_{i\in I}$, and let $P$ be an element of this ring.

Interpretations
For two interpretations of this definition, see:
 * Definition:Polynomial over Ring as Sequence
 * Definition:Polynomial over Ring as Function on Free Monoid on Set

Also see

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