Definition:Ring of Sequences

Definition
Let $\struct {R, +, \circ}$ be a non-null ring.

Given the Natural numbers $\N$, the ring of sequences over $R$ is the ring $\struct {R^{\N}, +', \circ'}$ induced on $R^{\N}$ by $+$ and $\circ$.

That is, the ring of sequences is the set of sequences in R with pointwise addition and pointwise product.

The ring operations on the ring of sequences over $R$ are defined:
 * $\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} +' \sequence {y_n} = \sequence {x_n + y_n}$
 * $\forall \sequence {x_n}, \sequence {y_n} \in R^{\N}: \sequence {x_n} \circ' \sequence {y_n} = \sequence {x_n \circ y_n}$

The zero of the ring of sequences is the sequence $\tuple{0,0,0,\dots}$, where $0$ is the zero in $R$.

By Structure Induced by Ring with Unity is a Ring with Unity, if $R$ is a ring with unity then the ring of sequences over $R$ is a ring with unity; namely the sequence $\tuple{1,1,1,\dots}$, where $1$ is the unity in $R$.

By Structure Induced by Commutative Ring is a Commutative Ring, if $R$ is a commutative ring then the ring of sequences over $R$ is a commutative ring.

Also known 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