Definition:Operations on Polynomial Ring of Sequences

Definition

Let $\struct {R, +, \circ}$ be a ring.

Let $P \sqbrk R$ be the set of all sequences in $R$:

$P \sqbrk R = \set {\sequence {r_0, r_1, r_2, \ldots} }$

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 $P \sqbrk R$ are defined as follows:

$(1)$	$:$	Ring Addition:	$\ds \sequence {r_0, r_1, r_2, \ldots} \oplus \sequence {s_0, s_1, s_2, \ldots} $	$\ds = $	$\ds \sequence {r_0 + s_0, r_1 + s_1, r_2 + s_2, \ldots} $
$(2)$	$:$	Ring Negative:	$\ds -\sequence {r_0, r_1, r_2, \ldots} $	$\ds = $	$\ds \sequence {-r_0, -r_1, -r_2, \ldots} $
$(3)$	$:$	Ring Product:	$\ds \sequence {r_0, r_1, r_2, \ldots} \odot \sequence {s_0, s_1, s_2, \ldots} $	$\ds = $	$\ds \sequence {t_0, t_1, t_2, \ldots} $	where $\ds t_i = \sum_{j \mathop + k \mathop = i} r_j \circ s_k$

\((1)\)	$:$	Ring Addition:	\(\ds \sequence {r_0, r_1, r_2, \ldots} \oplus \sequence {s_0, s_1, s_2, \ldots} \)	\(\ds = \)	\(\ds \sequence {r_0 + s_0, r_1 + s_1, r_2 + s_2, \ldots} \)
\((2)\)	$:$	Ring Negative:	\(\ds -\sequence {r_0, r_1, r_2, \ldots} \)	\(\ds = \)	\(\ds \sequence {-r_0, -r_1, -r_2, \ldots} \)
\((3)\)	$:$	Ring Product:	\(\ds \sequence {r_0, r_1, r_2, \ldots} \odot \sequence {s_0, s_1, s_2, \ldots} \)	\(\ds = \)	\(\ds \sequence {t_0, t_1, t_2, \ldots} \)	where $\ds t_i = \sum_{j \mathop + k \mathop = i} r_j \circ s_k$