Definition:Polynomial Ring

Definition
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.

Notation
It is common to denote a polynomial ring $(S, f, X)$ over $R$ as $R[X]$, where $X$ is the indeterminate of $(S, f, X)$.

The embedding $f$ is then implicit.

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 Evaluation Homomorphism
 * Definition:Ring of Polynomial Functions