\begin{definition}[Definition:Ring of Sequences]
Let $\struct {R, +, \circ}$ be a ring.
Given the natural numbers $\N$, the '''ring of sequences over $R$''' is the ring of mappings $\struct {R^\N, +', \circ'}$ where:
:$(1): \quad R^\N$ is the set of sequences in $R$
:$(2): \quad +'$ and $\circ'$ are the (pointwise) operations induced by $+$ and $\circ$.
From Structure Induced by Ring Operations is Ring, $\struct {R^\N, +', \circ'}$ is a ring.
\end{definition}