Definition:Operations on Polynomial Ring of Sequences

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

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

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

The operations ring addition $\oplus$, ring negative, and ring product $\odot$ on $P \left[{R}\right]$ are defined as follows:

Also see

 * Polynomial Ring of Sequences is Ring