Definition:Polynomial Ring/Sequences

Definition
Let $R$ be a commutative ring with unity.

Let $R^{(\N)}$ be the ring of sequences of finite support over $R$.

Let $f : R \to R^{(\N)}$ be the mapping defined as:
 * $f(r) = \left \langle {r, 0, 0, \ldots}\right \rangle$.

Let $X$ be the sequence $\left \langle {0, 1, 0, \ldots}\right \rangle$.

The polynomial ring over $R$ is the ordered triple $\left(R^{(\N)}, f, X \right)$.

Also see

 * Equivalence of Definitions of Polynomial Ring
 * Polynomial Ring of Sequences Satisfies Universal Property