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 $P \in R \sqbrk X$ be a polynomial

is a short way of saying:

Let $R \sqbrk X$ be a polynomial ring in one variable over $R$, call its variable $X$, and let $P$ be an element of this ring.

It is of the form:

$\ds \map P x = a_0 + a_1 x + a_2 x^2 + \cdots + a_{n - 1} x^{n - 1} + a_n x^n$

Multiple Variables

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.

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