\begin{definition}[Definition:Ring of Sequences/Units]
Let $\struct {R, +, \circ}$ be a ring with unity $1$.
Let $\struct {R^\N, +', \circ'}$ be the ring of sequences over $R$.
Let $\sequence {x_n}$ be a sequence over the set of units $U_R$ of $R$.
From Unit of Ring of Mappings iff Image is Subset of Ring Units:
:$\sequence {x_n}$ is a unit in the ring of sequences over $R$
and:
:the inverse of $\sequence {x_n}$ is the sequence defined by:
::$\sequence {x_n}^{-1} \in R^\N : \sequence {x_n}^{-1} = \sequence {x_n^{-1}}$
\end{definition}