Definition:Polynomial Ring

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:

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 defined as
It is common for an author to define the polynomial ring using a specific construction, and refer to other constructions as a polynomial ring. At we deliberately do not favor any construction. All the more so because at some point it becomes irrelevant.

It is also common to call any ring isomorphic to a polynomial ring a polynomial ring. For the precise meaning of this, see Ring Isomorphic to Polynomial Ring is 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
 * Definition:Polynomial Function
 * Definition:Polynomial Evaluation Homomorphism

Generalizations

 * Definition:Polynomial Algebra