Definition:Ring of Sequences/Units

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Definition

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 $R$.

$\sequence {x_n}$ is a unit in the ring of sequences over $R$

and:

the inverse of $\sequence {x_n}$ is the sequence defined by:
$\sequence {x_n}^{-1} \in R^\N : \sequence {x_n}^{-1} = \sequence {x_n^{-1}}$