Definition:Polynomial Ring

Single Indeterminate
Let $R$ be a commutative ring with unity.

The set of polynomials over $R$ can be made a ring, depending on the definition of a polynomial that is being used.

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

that can be defined in several ways:

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 see

 * Definition:Polynomial (Abstract Algebra)
 * Definition:Polynomial Algebra, an analogue for rings without unity.
 * Definition:Polynomial Evaluation Homomorphism
 * Definition:Ring of Polynomial Functions
 * Definition:Field of Rational Fractions