Definition:Ring of Sequences

Definition
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.

Also denoted as
It is usual to use the same symbols for the induced operations on the ring of sequences over $R$ as for the operations that induces them.

Also see

 * Structure Induced by Ring Operations is Ring


 * Structure Induced by Ring with Unity Operations is Ring with Unity


 * Structure Induced by Commutative Ring Operations is Commutative Ring


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


 * Definition:Ring of Mappings