Definition:Polynomial over Ring

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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 $P \in R \left[{X}\right]$ 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.


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 $P \in R \left[{\left\langle{X_i}\right\rangle_{i \mathop \in I} }\right]$ 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.


Interpretations

For two interpretations of this definition, see:


Also see