\begin{definition}[Definition:Operations on Polynomial Ring of Sequences]
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:
{{begin-axiom}}
{{axiom | n = 1
        | lc= '''Ring Addition:'''
        | ml= \sequence {r_0, r_1, r_2, \ldots} \oplus \sequence {s_0, s_1, s_2, \ldots}
        | mo= =
        | mr= \sequence {r_0 + s_0, r_1 + s_1, r_2 + s_2, \ldots}
        | c = 
}}
{{axiom | n = 2
        | lc= '''Ring Negative:'''
        | ml= -\sequence {r_0, r_1, r_2, \ldots}
        | mo= =
        | mr= \sequence {-r_0, -r_1, -r_2, \ldots}
        | c = 
}}
{{axiom | n = 3
        | lc= '''Ring Product:'''
        | ml= \sequence {r_0, r_1, r_2, \ldots} \odot \sequence {s_0, s_1, s_2, \ldots}
        | mo= =
        | mr= \sequence {t_0, t_1, t_2, \ldots}
        | rc= where $\ds t_i = \sum_{j \mathop + k \mathop = i} r_j \circ s_k$
}}
{{end-axiom}}
\end{definition}