Definition:Polynomial Ring/Sequences

Definition
Let $\left({R, +, \circ}\right)$ be a ring.

Let $P \left[{R}\right]$ be the set of all sequences in $R$ whose domain is $\N$:
 * $P \left[{R}\right] = \left\{{\left \langle {r_0, r_1, r_2, \ldots}\right \rangle}\right\}$

such that each $r_i \in R$, and all but a finite number of terms is zero.

Let the operations $\oplus$ and $\odot$ on $P \left[{R}\right]$ be defined as follows:

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

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

Then $\left( \left({P \left[{R}\right], \oplus, \odot}\right), f, X \right)$ is known as the polynomial ring over $R$.

Also denoted as
It is usual, on presentation of a polynomial ring such as this, to denote the operations $\oplus$ and $\odot$ as the same as those of their counterparts in the underlying ring $\left({R, +, \circ}\right)$.

However, as this stage of the development of the concepts it is wise to provide separate symbols.

Also see

 * Polynomial Ring of Sequences is Ring
 * Polynomial Ring of Sequences Satisfies Universal Property