\begin{definition}[Definition:Polynomial over Ring/Multiple Variables]
Let $R$ be a commutative ring with unity.
Let $I$ be a set.
A '''polynomial over $R$ in $I$ variables''' is an element of a polynomial ring in $I$ variables over $R$.
Thus:
:''Let $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial''
is a short way of saying:
: Let $R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ be a polynomial ring in $I$ variables over $R$, call its family of variables $\left\langle{X_i}\right\rangle_{i \mathop \in I}$, and let $P$ be an element of this ring.
\end{definition}