Definition:Polynomial Ring

In one Indeterminate
Let $R$ be a commutative ring with unity.

The set of polynomials over $R$ can be made a ring.

A polynomial ring in one variable is a certain pointed algebra over $R$, that is, an ordered triple $\left({S, \iota, X}\right)$ where:
 * $S$ is a commutative ring with unity
 * $\iota : R \to S$ is a unital ring homomorphism, called canonical embedding
 * $X$ is an element of $S$, called indeterminate

that can be defined in several ways:

In Multiple Indeterminates
Let $R$ be a commutative ring with unity.

Let $I$ be a set.

A polynomial ring in $I$ variables is a certain $I$-pointed algebra over $R$, that is, an ordered triple $\left({S, \iota, f}\right)$ where:
 * $S$ is a commutative ring with unity
 * $\iota : R \to S$ is a unital ring homomorphism, called canonical embedding
 * $f : S \to R$ is a family, whose image consists of indeterminates

that can be defined in several ways:

Equivalence of definitions
While, strictly speaking, the above definitions of polynomial ring do define different objects, they can be shown to be isomorphic in a strong sense.

See Equivalence of Definitions of Polynomial Ring.

Also known as
The polynomial ring in one indeterminate over $R$ is often referred to as the polynomial ring over $R$.

That is, if no reference is given to the number of indeterminates, it is assumed to be $1$.

Also see

 * Definition:Polynomial Algebra, an analogue for rings without unity.
 * Definition:Polynomial Evaluation Homomorphism
 * Definition:Ring of Polynomial Functions