Definition:Ring of Sequences/Units

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 $R$.

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:
 * $\sequence {x_n}^{-1} \in R^\N : \sequence {x_n}^{-1} = \sequence {x_n^{-1}}$

Also see

 * Structure Induced by Ring Operations is Ring


 * Unit of Ring of Mappings iff Image is Subset of Ring Units