Definition:Polynomial Ring/Sequences

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

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

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

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

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

That is:
 * $R^{\left({\N}\right)}$ is regarded as $R \sqbrk X$
 * $\iota : R \to R \sqbrk X$ is the canonical embedding
 * $X$ is the Indeterminate

In particular, for $r \in R$ and $n \in \N$:
 * $\map \iota r \odot X^n = \sequence {\underbrace{0,\ldots,0}_n,r,0,\ldots}$

is regarded as and written as:
 * $r X^n$

Also see

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